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Graduate Theses and Dissertations Graduate College

1-1-2009

Enhancement of engine simulation using LESturbulence modeling and advanced numericalschemesYuanhong LiIowa State University

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Recommended CitationLi, Yuanhong, "Enhancement of engine simulation using LES turbulence modeling and advanced numerical schemes" (2009).Graduate Theses and Dissertations. Paper 10693.http://lib.dr.iastate.edu/etd/10693

Enhancement of engine simulation using LES turbulence modeling and

advanced numerical schemes

by

Yuanhong Li

A dissertation submitted to the graduate faculty

in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

Major: Mechanical Engineering

Program of Study Committee:

Song-Charng Kong, Major Professor

Theodore J. Heindel

Shankar Subramaniam

Zhijian Wang

Hailiang Liu

Iowa State University

Ames, Iowa

2009

Copyright © Yuanhong Li, 2009. All rights reserved.

ii

DEDICATION

To my parents, Silai Li and Lianfeng Lin

iii

TABLE OF CONTENTS

LIST OF TABLES.................................................................................................................... v

LIST OF FIGURES ................................................................................................................. vi

NOMENCLATURE ................................................................................................................ xi

ACKNOWLEDGEMENTS................................................................................................... xxi

ABSTRACT......................................................................................................................... xxiii

1 INTRODUCTION ............................................................................................................. 1

1.1 Background............................................................................................................1

1.2 Motivation..............................................................................................................3

1.3 Objectives ..............................................................................................................5

2 LITERATURE REVIEW .................................................................................................. 7

2.1 Introduction............................................................................................................7

2.2 Turbulence Modeling.............................................................................................7

2.3 Spray Modeling....................................................................................................26

2.4 Combustion Modeling .........................................................................................47

2.5 Adaptive Grid Methods........................................................................................51

2.6 Parallel Computing ..............................................................................................53

2.7 KIVA Code ..........................................................................................................55

3 DIESEL ENGINE COMBUSTION MODELING.......................................................... 57

3.1 Introduction..........................................................................................................57

3.2 LES Turbulence Models ......................................................................................57

3.3 Combustion and Emissions Models.....................................................................62

3.4 Spray Combustion Modeling ...............................................................................65

iv

3.5 Diesel Engine Simulation ....................................................................................77

4 GASOLINE SPRAY MODELING USING ADAPTIVE MESH REFINEMENT ........ 87

4.1 Introduction..........................................................................................................87

4.2 Adaptive Mesh Refinement .................................................................................87

4.3 Parallelization ....................................................................................................102

4.4 Gasoline Spray Simulation Using AMR............................................................108

4.5 Gasoline Engine Simulation Using MPI-AMR .................................................127

5 CONCLUSIONS AND RECOMMENDATIONS........................................................ 133

5.1 Conclusions........................................................................................................133

5.2 Recommendations..............................................................................................134

BIBLIOGRAPHY................................................................................................................. 136

v

LIST OF TABLES

Table 2.1. Breakup regimes and breakup times of droplets (Arcoumanis et al., 1997).......... 36

Table 3.1. Simulation conditions for a non-evaporative spray. .............................................. 67

Table 3.2. Experimental conditions for model validation....................................................... 75

Table 3.3. Engine specifications and operating conditions..................................................... 78

Table 4.1. Comparison of the computer time for different mesh sizes................................. 111

Table 4.2. Simulation Conditions. ........................................................................................ 117

vi

LIST OF FIGURES

Figure 1.1. Schematic of in-cylinder processes in a diesel engine. .......................................... 2

Figure 2.1. Typical filters used in LES. (a) cut-off filter in spectral space; (b) box filter in

physical space; (c) Gaussian filter in physical space. .................................................... 15

Figure 2.2. Blob-method of primary breakup. ........................................................................ 28

Figure 2.3. 1-D approximation of internal nozzle flow with cavitation. ................................ 29

Figure 2.4. Primary breakup model of Nishimura and Assanis (2000). ................................. 33

Figure 2.5. (a) Inwardly opening pressure-swirl atomizer, (b) outwardly opening pressure-

swirl atomizer (Baumgarten, 2006). ............................................................................... 34

Figure 2.6. Schematic illustration of drop breakup mechanisms (a) KH-type; (b) RT-type

(Kong et al., 1999). ......................................................................................................... 39

Figure 2.7. Combined blob-KH/RT model (Kong et al., 1999).............................................. 41

Figure 2.8. Regimes and mechanisms of droplet collision (Baumgarten, 2006). ................... 42

Figure 3.1. Three mesh sizes used in non-evaporative spray simulations in a constant volume

chamber (top view, injector is located at the center). ..................................................... 68

Figure 3.2. Spray liquid penetrations using different grid sizes in LES simulations.............. 69

Figure 3.3. Spray liquid penetrations using different grid sizes in RANS simulations. ......... 70

Figure 3.4. Spray structure using different grid sizes in LES simulations at 1.8 ms

(measurement by Dan et al., 1997). ................................................................................ 70

Figure 3.5. Spray structure using different grid sizes in RANS simulations at 1.8 ms

(measurement by Dan et al., 1997). ................................................................................ 71

vii

Figure 3.6. Velocity vector on a central cutplane (y=0) in both RANS and LES simulations at

1.8 ms.............................................................................................................................. 73

Figure 3.7. Vortex structure plotted as an iso-surface of the critical value cQ of 1.0e+6 in both

RANS and LES simulations at 1.8 ms. ........................................................................... 73

Figure 3.8. Comparison of soot distributions at 1.3 and 1.7 ms after start of injection for

dnozzle=100 µm, Tamb=1000 K, ΔPamb=138 MPa, ρamb=14.8 kg/m3 (measurement by

Pickett and Siebers, 2004)............................................................................................... 76

Figure 3.9. Comparison of soot iso-surface predicted by RANS and LES and the

experimental PLII images (Pickett and Siebers, 2004) for dnozzle=100 µm, Tamb=1000 K,

ΔPamb=138 MPa, ρamb=14.8 kg/m3................................................................................. 76

Figure 3.10. Comparison of predicted temperature distributions using RANS and LES for

dnozzle=100 µm, Tamb=1000 K, ΔPamb=138 MPa, ρamb=14.8 kg/m3................................ 77

Figure 3.11. Computational mesh (a 60° sector mesh)........................................................... 79

Figure 3.12. Comparisons of measured (solid lines) (Klingbeil et al., 2003) and predicted

(dashed) cylinder pressure and heat release rate data for SOI = –20, –10 and +5, both

with 8% EGR. ................................................................................................................. 81

Figure 3.13. Evolutions of predicted in-cylinder soot and NOx emissions compared with

engine exhaust measurements (solid symbols) (Klingbeil et al., 2003) for SOI = –5

ATDC, 40% EGR. .......................................................................................................... 82

Figure 3.14. Comparisons of measured (Klingbeil et al., 2003) and predicted soot and NOx

emissions for 8% EGR cases. ......................................................................................... 83

viii

Figure 3.15. Comparisons of measured (Klingbeil et al., 2003) and predicted soot and NOx

emissions for 27% EGR cases. ....................................................................................... 83

Figure 3.16. Comparisons of measured (Klingbeil et al., 2003) and predicted soot and NOx

emissions for 40% EGR cases. ....................................................................................... 84

Figure 3.17. Spray and temperature distributions on cross sections through spray at –14

ATDC for SOI = –20 ATDC with 8% EGR. .................................................................. 85

Figure 3.18. Distributions of CO mass fraction on a cross section through spray at –14 ATDC

for SOI = –20 ATDC with 8% EGR............................................................................... 86

Figure 4.1. Schematic of local mesh refinement: (a) a parent cell is divided into eight child

cells; (b) conventional face interface (1-3) and coarse-fine interface (1-2).................... 89

Figure 4.2. Geometric arrangement of the gradient calculation at an interface f. .................. 94

Figure 4.3. Computational stencil for gradient calculation................................................... 100

Figure 4.4. Domain partition for four processors using METIS........................................... 105

Figure 4.5. Comparison of liquid and vapor penetrations using different meshes with

respective refinement levels.......................................................................................... 111

Figure 4.6. Comparison of the maximum gas velocity for different mesh refinement

conditions...................................................................................................................... 112

Figure 4.7. Total active cell numbers as a function of crank angle for different mesh

refinement conditions.................................................................................................... 113

Figure 4.8. Total cell numbers as a function of crank angle for different mesh refinement

conditions...................................................................................................................... 113

Figure 4.9. Comparison of the spray pattern and fuel vapor on a cutplane along the spray axis

at 205 ATDC................................................................................................................. 114

ix

Figure 4.10. Comparison of the spray pattern on a cutplane along the spray axis (top) and at

3.0 cm above the piston surface (bottom) at 205 ATDC. ............................................. 115

Figure 4.11. Comparison of the spray pattern and fuel vapor for injection with a slanted

angle.............................................................................................................................. 116

Figure 4.12. Comparison of liquid penetrations for different numbers of processors.......... 118

Figure 4.13. Comparison of spray patterns at crank angle 205 ATDC using different numbers

of processors together with DAMR. ............................................................................. 119

Figure 4.14. Liquid penetration comparison with the 10 timestep control strategy. ............ 120

Figure 4.15. Spray pattern comparison between the baseline serial run and parallel runs with

the 10-timestep control strategy.................................................................................... 120

Figure 4.16. Spray pattern comparisons in the DI gasoline engine combustion chamber at

crank angle 310 ATDC. Note that the start of injection was 180 ATDC. .................... 121

Figure 4.17. Speed-up and computational efficiency of the three geometries for single-hole

injection using the coarse mesh. ................................................................................... 123

Figure 4.18. Comparisons of speed-up and computational efficiency for single-hole and six-

hole injections using the coarse mesh........................................................................... 124

Figure 4.19. Comparisons of speed-up and computational efficiency between coarse mesh

and fine mesh for using single-hole nozzle................................................................... 126

Figure 4.20. 3-D meshes of a 4-valve pent-roof engine and a 4-vertical-valve engine. ....... 127

Figure 4.21. Comparisons of fuel vapor mass fraction and temperature for 4-valve pent-roof

engine with and without AMR at 35 ATDC (SOI=10 ATDC)..................................... 128

Figure 4.22. Parallel performance for 4-valve pent-roof engine with and without AMR

(SOI=10 ATDC). .......................................................................................................... 130

x

Figure 4.23. Comparisons of fuel vapor mass fraction and spray for 4-vertical-valve engine at

485 ATDC (SOI=460ATDC). ...................................................................................... 130

Figure 4.24. Speed-up for 4-vertical-valve engine with and without AMR (SOI=460ATDC).

....................................................................................................................................... 131

xi

NOMENCLATURE

Roman Symbols

a acceleration or a pressure gradient scaling parameter for low Mach flow

1,2 4,3 ,, , c cna a a geometric coefficients

fA face area vector

0A a switch for turbulent or laminar flow

sfA constant

0B constant

lc liquid specific heat

1, 2 3,C C C model coefficients or constants

, , ,b d e FC C C C constants

Dc drag coefficient

mC constant

1, 2C Cε ε turbulent model constants

, , ,k sC C C Cε μ constants

pc specific heat at constant pressure

1 2,t tC C coefficients for turbulent dissipation rate equation

vc specific heat at constant volume

d droplet diamter

xii

, ,l iD D binary diffusion coefficients or material derivative

aD Damkohler number

cD gas diffusion coefficient

sD average particle diameter

e specific internal energy

E computational efficiency

f probability density function

sjF rate of momentum gain per unit volume due to spray

g gravity vector

G filter function at the grid level

TG filter function at the test level

, ,i j k unit vectors in x,y, z coordinate directions, respectively

I specific internal energy

k turbulent kinetic energy

1k a constant

gK thermal conductivity

L latent heat of vaporization

AL turbulence length scale

bL breakup length

ijL Leonard stress tensor

xiii

2 2C HM mass of acetylene

ijM a stress tensor

, ,s sf soM M M masses of soot, soot formation and soot oxidation

cMw soot molecular weight

n̂ outward unit normal vector

N total number of species or cells or processors

scn number of convective sub-cycles

dNu Nusselt number

,,s s lh h enthalpy

p thermodynamic pressure

kP production term in turbulent kinetic energy equation

Pr turbulent Prandtl number

q molecular heat flux by conduction or generic cell-centered variable

Q generic cell-centered variable

cQ source term in energy equation due to chemical reactions

dQ rate of heat transfer to drop surface per unit area by conduction

sQ source term in energy equation due to spray vaporization

TQ general source term in energy equation

r droplet radius in spherical shape

R universal gas constant

xiv

critr critical value for mesh refinement and coarsening

Re Reynolds number

mr ratio of the total mass of liquid and vapor in a cell to total injected fuel mass

TotalR surface mass oxidation rate

s stoichiometric coefficient for a fuel

S speedup

Sc turbulent Schmidt number

ijS rate of strain tensor

S magnitude of the rate of strain tensor

St Stokes number

t time

T Taylor number or clock time

ijT ‘test’ level stress tensor

u velocity

*u modeled instantaneous velocity

V computational cell volume

sW rate of work done by turbulent eddies to disperse spray droplets

gWe Weber number

, ,x y z coordinate direction

y normalized droplet distortion

lY species mass fraction

xv

z mixture fraction

Z Ohnesorg number

Greek Symbols

,α β constants for mesh refinement and coarsening

χ scalar dissipation rate or a distribution

Δ grid filter length

Δ test filter length

ijδ Kronecker delta function

fVδ face volume change

, ct tΔ Δ a main computational timestep and a subcycle timestep

ε turbulent dissipation rate

lijε the alternating tensor

φ a generic variable

κΔ cut-off wave number

λ thermal conductivity

Λ wave length

μ molecular or effective viscosity

tμ turbulent viscosity

∇ gradient operator

ν molecular kinematic viscosity

tν turbulent kinematic viscosity

xvi

ω rate of chemical reaction

ω turbulence frequency

Ω wave growth rate

ρ thermodynamic density

sρ soot density

φ a variable

π a constant (=3.1415926)

σ surface tension force

ijσ viscous stress tensor

,k εσ σ model constants

θ combustion progress variable

τ standard deviation

Aτ turbulence time scale

,c jτ indicator for mesh refinement and coarsening

collτ turbulence correlation time

breakτ droplet breakup time

burstτ bubble burst time

eτ eddy life time

si iF uτ subgrid droplet-gas interaction

ijτ subgrid scale stress tensor

rτ droplet aerodynamic response time

xvii

i lu Yτ subgrid turbulent species flux

iu Tτ subgrid turbulent energy flux

Subscripts and superscripts

c source term due to chemistry or child cell

drop spray droplet

eff effective

f a face

F fuel

g gas phase

h higher level child cell

hole nozzle hole

inj fuel injection

, ,i j k indices for Cartesian coordinates

l index for species or liquid phase or lower level cell

0,L L Lagrangian phase at time n+1 and n, respectively

O oxidizer

p parent cell

P product

rel relative

s source term due to spray or a surface

st stoichiometric

T matrix transpose

xviii

vap vapor

Other Symbols

mean quantity of ensemble average or resolved scale of LES filtering in

incompressible flows

mean quantity of Favre ensemble average or resolved scale of Favre filtered

quantity in compressible flows

two times ensemble average or LES filtering in incompressible flows

two times Favre ensemble average or Favre LES filtering in compressible

flows

three times Favre LES filtering in compressible flows

filtered quantity at the test filter level

' fluctuation with respect to ensemble average, or subgrid scale component of

filtered quantity

" fluctuation with respect to Favre ensemble average, or subgrid scale

component of Favre filtered quantity

Abbreviations

AMR adaptive mesh refinement

ATDC after top dead center

BDC bottom dead center

CAD crank angle degree

CFD computational fluid dynamics

CHEMKIN a chemistry solver that solves complex chemical kinetics problems

xix

CO carbon monoxide

DI direct injection

DNS direct numerical simulation

EGR exhaust gas recirculation

EVO exhaust valve open

ICEM-CFD a commercial package for computational grid generation

IVC intake valve close

KIVA a computer code developed at Los Alamos National Laboratory

KIVA-3V version 3 of KIVA computer code

KIVA-4 version 4 of KIVA computer code

LES,VLES large eddy simulation, and very large eddy simulation, respectively

MPI Message Passing Interface

METIS a software package for parallel domain decomposition based on a multilevel

recursive-bisection scheme

h METIS a software package for parallel domain decomposition based on multilevel

hypergraph partitioning schemes

NO nitric oxide

2NO nitrogen dioxide

2N O nitrous oxide

xNO nitrogen oxides (NO+NO2)

NS Navier-Stokes

PDF probability density function

xx

PLII planar laser induced incandescence

RANS Reynolds Averaged Navier-Stokes

RNG Renormalization group

RPM reciprocation per minute

RSM Reynolds stress models

SGS subgrid scale

SOI start of injection

TAB Taylor analogy breakup

TDC top dead center

TKE turbulent kinetic energy

xxi

ACKNOWLEDGEMENTS

First and foremost, I would like to express my sincerest gratitude to my advisor, Professor

Song-Charng Kong, for accepting me as his PhD student and for his guidance,

encouragement, patience, and numerous other helps in various aspects. He not only

broadened my knowledge of numerical simulation, combustion, and IC engine but also

inspired me to overcome so many obstacles in my research. His thorough reviews on my

papers and thesis also helped me greatly improve my writing skills. I am grateful to other

committee members for their efforts and contributions to this work. Professor Theodore J.

Heindel patiently read through my thesis manuscript and provided very detailed and critical

comments on the improvement of my thesis. Professor Shankar Subramaniam provided me

with useful knowledge of multiphase flow modeling and multi-scale modeling. Professor

Zhijian Wang taught me various advanced numerical methods in CFD and provided

insightful comments on the numerical schemes for adaptive mesh refinement. Professor

Hailiang Liu served as representative of my Master minor from Mathematics Department and

also taught me a great deal of knowledge about numerical analysis. Finally, I would like to

thank Professor Richard H. Pletcher for accepting me into the PhD program in Mechanical

Engineering and leading me into the field of LES turbulence modeling. Without their help,

this work would not have been possible.

I am indebted to Dr. David J. Torres at Los Alamos National Laboratory for providing me

important comments and help in the development of parallel AMR into KIVA-4. I would like

xxii

to thank other faculty, staff, and so many graduate students in Mechanical Engineering

Department for their help throughout my study at ISU.

I am deeply indebted to my parents for their continual and strong support. I also thank other

family members for their encouragement and great support.

This research was partially supported by Ames Laboratory, U.S. Department of Energy, and

Ford Motor Company. The computer resources provided by the Iowa State High

Performance Computing Group are also gratefully acknowledged.

xxiii

ABSTRACT

The goal of this study is to develop advanced numerical models and algorithms to improve

the accuracy of engine spray combustion simulation. This study developed a large eddy

simulation (LES) turbulence model and adaptive mesh refinement (AMR) algorithms to

enhance the accuracy and computational efficiency of engine simulation.

The LES approach for turbulence modeling is advantageous over the traditional Reynolds

Averaged Navier Stokes (RANS) approach due to its capability to obtain more detailed flow

information by resolving large-scale structures which are strongly geometry dependent. The

current LES approach used a one-equation, non-viscosity, dynamic structure model for the

sub-grid stress tensor and also used a gradient method for the sub-grid scalar fluxes. The

LES implementation was validated by comparing the predicted spray penetrations and

structures in a non-evaporating diesel spray. The present LES model, when coupled with

spray breakup and detailed chemistry models, were able to predict the overall cylinder

pressure history, heat release rate data, and the trends of NOx and soot emissions with respect

to different injection timings and EGR levels in a heavy-duty diesel engine. Results also

indicated that the LES model could predict the unsteadiness of in-cylinder flows and have the

potential to provide more detailed flow structures compared to the RANS model.

AMR algorithms were also developed to improve transient engine spray simulation. It is

known that inadequate spatial resolution can cause inaccuracy in spray simulation using the

stochastic Lagrangian particle approach due to the over-estimated diffusion and inappropriate

xxiv

liquid-gas phase coupling. Dynamic local mesh refinement, adaptive to fuel spray and vapor

gradients, was developed to increase the grid resolution in the spray region. AMR was

parallelized using the MPI library and various strategies were also adopted in order to

improve the computational efficiency, including timestep control, reduction in search of the

neighboring cells on the processor boundaries, and re-initialization of data at each adaptation.

The AMR implementation was validated by comparing the predicted spray penetrations and

structures. It was found that a coarse mesh using AMR could produce the same results as

those using a uniformly fine mesh with substantially reduced computer time. The parallel

performance using AMR varied depending on the geometry and simulation conditions. In

general, the computations without valve motion or using a fine mesh could obtain better

parallel performance than those with valve motion or using a coarse mesh.

1

1 INTRODUCTION

1.1 Background

Internal combustion engines will continue to be the main power source in the foreseeable

future for transportation, off-road machinery, ships, and many other applications despite that

alternative power sources are being developed. Internal combustion engines have been faced

with tremendous challenges in meeting increasingly stringent emissions regulations and

requirements for power density and fuel efficiency. These challenges have drawn extensive

interests in exploring various ways to improve combustion efficiency and reduce pollutant

emissions. Various strategies such as multiple fuel injections, low temperature combustion,

homogeneous charge compression ignition (HCCI) operation, alternative fuels, and exhaust

gas aftertreatment are investigated to achieve more efficient combustion and lower pollutant

emissions. Both experiments and modeling are necessary tools to conduct the investigations.

Numerical simulation can provide results faster and cheaper, and can give extensive

information about the complex in-cylinder processes as shown in Figure 1.1. Numerical

simulation can also be used to investigate processes that take place at time and length scales

or in places that are not accessible using experimental techniques. Therefore, numerical

simulation has become a necessary complement to experiments in developing advanced

engine technologies. Today, the complex task of developing advanced mixture formation

and combustion strategies can only be achieved with a combination of experimental and

numerical studies

2

Intake port Exhaust portInjector

Comb. chamber

vapor

Piston

flame

Heattransfer

nozzle

Gas flowspray

Figure 1.1. Schematic of in-cylinder processes in a diesel engine.

Three classes of models have been used to perform numerical simulation of internal

combustion engines ranging from zero-dimensional thermodynamic models, one-dimensional

phenomenological models to three-dimensional computational fluid dynamics (CFD) models

in the order of increasing complexity. The thermodynamic model describes only the most

relevant processes without considering any spatial resolution and cannot provide insight into

the local flow details. The phenomenological model considers quasi-spatial resolution of the

combustion chamber and uses appropriate sub-models to approximate the relevant processes

such as mixture formation, ignition and combustion. This model may be used to predict the

global heat release rate and overall engine performance. The CFD model solves the

discretized set of Navier-Stokes equations in combination with an appropriate turbulence

model, wall heat transfer models, and spray combustion models for chemically reactive flows

3

with sprays. This model takes into account every process of interest. For instance, the

processes of fuel injection, liquid breakup and drop evaporation, collisions, and wall

impingement are simulated using various submodels. These models can provide detailed

information of the processes in time and space. This rich information makes the CFD model

particularly suited for investigating the effects of the in-cylinder processes on engine

performance. Compared to the other two simpler models, the CFD model consumes

significantly more computer time and memory. It is necessary to develop high-fidelity

models with high computational efficiency.

1.2 Motivation

Appropriate mixture preparation is critical for achieving efficient combustion and low

emissions, especially in direct-injection engines. Mixture formation is determined by the

interaction of the spray and the in-cylinder flow, which in turn is determined by the intake

port and combustion chamber geometry. Accurate modeling of mixture distribution requires

appropriate simulation of the flow field. Large eddy simulation (LES) turbulence modeling

can offer improved simulation of the in-cylinder flow process and is a promising approach

for more accurate engine simulation.

In the LES modeling, all the scales larger than the computational grid size are directly solved

by a space- and time-accurate method while the effects of scales below the grid size are

modeled using sub-grid models. LES requires models for sub-grid scale physical processes

that occur below the grid resolution. An important example includes the chemical process of

4

pollutant formation. Traditionally, LES has focused only on modeling the turbulent sub-grid

stresses. However, engine applications are unique in the turbulence community and require a

much broader range of sub-grid models including scalar mixing, spray dynamics, drop

vaporization, combustion, and emissions. For instance, sub-grid modeling of the scalar

dissipation rate plays an important role in predicting mixing since the scalar dissipation is

used for representing sub-grid distributions of scalar fields in the probability density function

modeling and also the sink term in the scalar variance transport equation. The LES filtering

also gives rise to the unclosed terms due to engine spray and combustion chemistry, which

still pose modeling difficulties. Thus, research is needed to develop an adequate LES

turbulence model that can be used for engine simulation.

On the other hand, the most common spray description is the Lagrangian discrete particle

method based on Dukowicz’s technique (Dukowicz, 1980), in which the properties of

representative droplet parcels are randomly chosen from theoretical distribution functions.

The continuous gaseous phase is described by an Eulerian method. Spray-gas interactions

are achieved by the coupling of source terms for the exchange of mass, momentum, and

energy. Numerical studies have indicated that spray simulation using the stochastic

Lagrangian discrete method is sensitive to the grid resolution and grid arrangement

(Abraham, 1997; Aneja and Abraham, 1998; Subramaniam and O’Rourke, 1998; Beard et al.,

2000; Schmidt and Rutland, 2000; Hieber, 2001; Are et al., 2005; Tonini et al., 2008). This

grid dependence can be partially attributed to the inadequate spatial resolution of the

coupling between the gas and liquid phases. In a region near the injector exit, velocity and

species density gradients can be strong when a high-velocity spray is injected into a slowly

5

moving gas. Inadequate spatial resolution of the spray in this region can result in a much

higher relative velocity between the two phases and an over-estimation of diffusion (Stiesch,

2003; Baumgarten, 2006). Thus, it is often required to use a fine mesh in the region near the

nozzle exit to improve spray modeling while using a coarser mesh away from the nozzle exit

to save computer time.

The grid dependence has also been shown to be a result of the sub-models for spray

simulations (Aneja and Abraham, 1998; Schmidt and Rutland, 2000; Hieber, 2001; Are et al.,

2005). While improving the sub-models to reduce the grid dependence has attracted much

attention (Hieber, 2001; Schmidt and Rutland, 2004; Are et al., 2005; Tonini et al., 2008), the

improvements may still be subject to further validation and tuning of model constants for

specific conditions. Another way to reduce grid dependence is to increase the local spatial

resolution in the spray region. Adaptive mesh refinement (AMR) (Bell et al., 1994a; Wang

and Chen, 2002; Anderson et al., 2004; Lippert et al., 2005) can be utilized to increase the

spatial resolution in the spray region to improve the phase coupling and thus alleviate the

mesh dependence without incurring much computer time. In particular, AMR can provide a

greater flexibility of adapting to highly transient spray with adequate spatial resolution that

otherwise would not be readily available if a uniform mesh is used.

1.3 Objectives

The objectives of this study are twofold. The first objective is to develop large eddy

simulation (LES) to improve modeling of in-cylinder flow and combustion in diesel engines.

6

The second objective is to develop parallel adaptive mesh refinement algorithms to increase

the spatial resolution in the spray region to improve the accuracy and computational

efficiency of spray modeling.

7

2 LITERATURE REVIEW

2.1 Introduction

CFD models have been widely used for three-dimensional simulation of fluid flow, mixture

formation, combustion, and pollutant formation in combustion systems. This chapter presents

a review of the physical and chemistry models that are relevant to engine simulation. These

models include those for turbulence, spray, combustion, and emissions. Additionally,

numerical schemes for adaptive mesh refinement and parallel computing are also reviewed.

2.2 Turbulence Modeling

Turbulent flows in internal combustion engines are characterized by a wide range of length

and time scales (Reynolds, 1980). The length scale varies from the smallest Kolmogorov

scale to the largest integral scale comparable to the engine geometry. Widely varying time

scales exist and are associated with different flow structures such as homogeneous flows,

shear layers, boundary layers, and re-circulations (Reynolds, 1980). Depending on the

resolution of the turbulence energy spectrum, turbulent modeling can be divided into three

categories: Reynolds Averaged Navier-Stokes (RANS) approach, large eddy simulation

(LES), and direct numerical simulation (DNS). The RANS approach only resolves the mean

flow fields whereas models the turbulence energy, and it is computationally more efficient

than the other two approaches. The LES approach resolves the large scales of the turbulence

at the level of the computational grid size and requires sub-grid models to model the effects

of smaller scales. The DNS approach resolves the smallest eddies in a flow field without

8

averaging or using turbulence models, and it is computationally expensive. DNS simulations

have been limited to basic research applications with low Reynolds numbers and small

geometric domains. The fundamentals of governing equations of RANS and LES approaches

will be presented as follows. The instantaneous conservation equations for mass, species,

momentum and enthalpy in a turbulent reacting flow are expressed as (Kuo, 1986)

( )

0j

j

ut x

ρρ ∂∂+ =

∂ ∂ (2.1)

( ) ( ) ( ) ( ), 1,j l l j lll

j j

u Y D YYl N

t x xρρ

ω∂ ∂∂

+ = − + =∂ ∂ ∂

(2.2)

( )i j iji

j i j

u uu pt x x x

ρ σρ ∂ ∂∂ ∂+ = − +

∂ ∂ ∂ ∂ (2.3)

( ) ( ), ,

1

Nj ss i

j l j l s l ij Tlj j j j j j

u hh up p Tu D Y h Qt x t x x x x x

ρρλ ρ σ

=

∂ ⎛ ⎞∂ ∂∂ ∂ ∂ ∂ ∂ ⎛ ⎞+ = + + − + +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

∑ (2.4)

where ρ is gas density, lY is mass fraction for species l , u is velocity, ,l jD is the binary

mass diffusion coefficients, lω is chemical reaction rate for species l , p is static pressure,

ijσ is the rate of stress tensor, sh is total specific enthalpy, λ is the thermal conductivity, x

is coordinate, and t is time. In Eq. (2.4), on the right hand side, the first two terms represent

the material derivative of the pressure, the third and fourth terms represent heat flux due to

conduction and enthalpy diffusion, respectively, the fifth term is viscous dissipation, and the

last term is a source term. The source term TQ in the energy equation accounts for the

contributions from the heat release from combustion, and heat transfer between the liquid

droplets and the gas phase. The heat transfer due to radiation is generally neglected in engine

9

modeling due to the following reasons. First, the combustion duration (e.g., 40~50 CAD) is

relatively short compared to the entire engine cycle (i.e., 720 CAD). During diesel

combustion at high-load conditions, luminous flame due to soot radiation generally only lasts

for 20 to 30 CAD, and the effects of radiative heat transfer on total wall heat loss was not

significant, particularly in part-load conditions (Wiedenhoefer and Reitz, 2003). There were

only very limited studies that considered radiative heat transfer in engine simulation partly

due to the complexity of the integro-differential equations associated with thermal radiation

(Libby and Williams, 1980; Coelho, 2007).

RANS approach

Most turbulence model development has been based on the RANS approach. Following the

basic approach of Reynolds, the instantaneous value of the turbulent flow quantity φ is split

into a mean φ and a fluctuating component 'φ

'φ φ φ= + . (2.5)

φ is an ensemble-averaged quantity, e.g., 1

1 M

iiM

φ φ=

= ∑ . For statistically steady flows, it is

the time-averaged and defined by

( )1 ,t t

tx t dt

tφ φ

+Δ=Δ ∫ . (2.6)

For compressible flows, mass-weighted averages or Favre averages are often introduced to

avoid unclosed correlations between any quantity φ and density fluctuations, and the

situation of unconserved mass resulting from the above Reynolds averages (Kuo, 1986). The

Favre average is defined as

10

____

ρφφρ

= . (2.7)

The Favre-average is used for all flow quantities except for pressure and density whose

averages are obtained using the Reynolds average. Then a quantity φ can be expressed as its

Favre average φ and fluctuating component ''φ as

___

'' '' with 0φ φ φ φ= + = (2.8)

Substituting these decomposition terms into the original Navier-Stokes equations yields the

RANS equations as follows.

( )

0j

j

ut x

ρρ ∂∂+ =

∂ ∂ (2.9)

( ) ( ) ( )

_______'' ''

, 1,l j ll j l j l l

j j

Y u YD Y u Y l N

t x x

ρ ρρ ω

∂ ∂ ∂ ⎛ ⎞+ = − + + =⎜ ⎟∂ ∂ ∂ ⎝ ⎠ (2.10)

( ) ( )'' ''i ji

ij i jj i j

u uu p u ut x x x

ρρ σ ρ∂∂ ∂ ∂

+ = − + −∂ ∂ ∂ ∂

(2.11)

( ) ( ) '' '', ,

1

Ns j s ij j s l j l s l ij T

lj j j j j j

h u h up p Tu u h D Y h Qt x t x x x x x

ρ ρλ ρ ρ σ

=

∂ ∂ ⎛ ⎞ ⎛ ⎞ ∂∂ ∂ ∂ ∂ ∂+ = + + − − + +⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

∑

(2.12)

where

''j j j

j j j

p p pu u ux x x∂ ∂ ∂

= +∂ ∂ ∂

(2.13)

Some terms in the above equations are the results of Favre-averaging and highly nonlinear

and require additional closure terms. The most famous terms among them are Reynolds

11

stresses '' ''i ju u , species turbulent flux '' ''

j lu Y and enthalpy turbulent flux '' ''j su h , and chemical

reaction rates lω .

The Reynolds stresses '' ''i ju u must be closed by a turbulence model. Boussinesq (1877)

assumed that the apparent turbulent shearing stresses might be related to the rate of mean

strain through an apparent scalar turbulent viscosity (Tennekes and Lumley, 1972; Hinze,

1975; Tannehill et al., 1997). Turbulence models can fall into two categories depending on

whether the Boussinesq assumption was applied. The models using this assumption are

called turbulent viscosity models or first-order models. Models without using this assumption

are called Reynolds stress models (RSM) or second-order closures.

Following the Boussinesq assumption, the Reynolds stresses are generally modeled using the

constitutive relation for Newtonian fluids

'' '' 23i j t ij iju u S kν δ= − + (2.14)

where /t tν μ ρ= is the eddy viscosity, ijS is the rate of strain tensor,

12

jiij

j i

uuSx x

⎛ ⎞∂∂= +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

, (2.15)

ijδ is the Kronecker delta function. k is the turbulent kinetic energy and defined as

" "12 i ik u u= (2.16)

12

The eddy viscosity tν is unknown and requires a closure. Typical approaches to close this

term include algebraic or zero-equation expressions, one-equation closure, and two-equations

closure.

A typical zero-equation model is the Prandtl mixing length model. In this model, the scalar

eddy viscosity is linked to the velocity gradient via an algebraic expression. Prandtl and

Kolmogorov (1940s) suggested that the characteristic velocity of turbulence be proportional

to the square root of the turbulent kinetic energy, which leads to so-called one-equation

model.

The most popular turbulence models in RANS approach are the class of two-equation models.

Among them, the k ε− model is the most widely used due to its simplicity and cost

effectiveness. This model allows the characteristic length scale cl to vary depending on the

flow. The scalar eddy viscosity is modeled using the length scale determined from the

turbulent kinetic energy and its dissipation rate (Jones and Launder, 1972) as

2

tkCμνε

= . (2.17)

The turbulent kinetic energy k and its dissipation rate ε are described by two transport

equations as follows

( ) ( )j tk

j j k j

u kk k Pt x x x

ρρ μμ ρεσ

∂ ⎡ ⎤∂ ⎛ ⎞∂ ∂+ = + + −⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

, (2.18)

( ) ( ) 2

1 2j t

kj j j

u kC P Ct x x x kε ε

ε

ρ ερε μ ε εμ ρσ ε

∂ ⎡ ⎤∂ ⎛ ⎞∂ ∂+ = + + −⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

(2.19)

13

where kP is the production term and is given by

" " ik i j

j

uP u ux

ρ ∂= −

∂ (2.20)

Standard values of the model constants are also used in engine calculations (Amsden et al.,

1989) since these values are chosen based on analytical solutions to the model in order to

meet certain experimental constraints (Durbin and Reif, 2001). The standard values are given

as

1 20.09; 1.0; 1.3; 1.44; 1.92kC C Cμ ε ε εσ σ= = = = = . (2.21)

This model can readily provide turbulent time and length scale estimates that can be used in

turbulent combustion models. Other two-equation models have been developed using

various combinations of k and ε to derive alternative transport equations to replace the

equations for ε . For instance, the k ω− solves the transport equation for turbulence

frequency / kω ε= in place of the dissipation rate (Wilcox, 1993). One advantage of the

k ω− model over the k ε− model is its treatment of the near-wall region in boundary layer

flows, especially for low Reynolds number flows (Wilcox, 1993).

Turbulence models based on the turbulent viscosity hypothesis may not perform well in

complex flows, i.e., flows with significant streamline curvature or mean pressure gradients.

The general k ε− model assumes isotropic turbulence but practical flows often exhibit

strong anisotropic features. Corrections based on non-linear turbulent viscosity models have

proven to be useful for predicting secondary flows. But their accuracy still needs to be

verified in predicting more complex flows such as with strong mean streamline curvature

and/or rapid changes in the mean velocity (Pope, 2000). More sophisticated models have also

14

been proposed such as the second order Reynolds stress models (RSM) (Launder, 1996).

Nonetheless, the traditional two-equation k ε− models are still widely used in engine

applications.

LES approach

In turbulent flows, energy-containing scales (the largest scales) determine most of the flow-

dependent transport properties while energy-dissipating scales (the smallest scales) exhibit

more universal and isotropic characteristics (Hinze, 1975; Pope, 2000; Durbin and Reif, 2001;

Fox, 2003). It is also observed that the energy-containing scales can be modeled

independently of the universal energy-dissipating scales given a model for the flux of energy

through the inertial range. Therefore, the objective of the large eddy simulation approach is

to explicitly resolve the largest scales of the flow field whereas the effects of the smallest

ones are modeled. At high enough Reynolds number, it is expected that the small scales will

be flow-independent, and thus that they can be successfully modeled by an appropriate

subgrid scale (SGS) model.

LES is based on the decomposition of a flow variable into resolved and unresolved (sub-grid

scale) terms. For any flow variable φ , it can be decomposed as

'φ φ φ= + (2.22)

where φ is the filtered quantity (or resolved scale) and 'φ is the sub-grid scale (or unresolved

scale) quantity. Note that the filtered quantity φ is different than the Reynolds averaged φ

15

in the RANS approach which is an ensemble averaged value. The filtered quantity φ is

defined through the use of a filtering function in spectral space or in physical space as

( ), ( ) ( , )V

t G t dφ φ= −∫x x y y y (2.23)

where ( )G −x y is the filter function which should be smooth, rotationally symmetric, and

satisfy the normalization condition

( ) 1V

G d =∫ y y . (2.24)

Figure 2.1. Typical filters used in LES. (a) cut-off filter in spectral space; (b) box filter in

physical space; (c) Gaussian filter in physical space.

Commonly used spatial filter functions include wave number cut-off filter, ‘box’ filter, and

Gaussian filter (Poinsot and Veynante, 2001; Berselli et al., 2006; Sagaut, 2006) as shown in

Figure 2.1. The wave number cut-off filter is used for LES in spectral space and is defined

as

( )1 / ,( )

0 otherwise.k

G kκ πΔ⎧ ≤ = Δ

= ⎨⎩

(2.25)

For this filter, all wave numbers below a cut-off, κΔ , are resolved, while all wave numbers

above the cut-off are modeled.

16

The ‘box’ filter is given by

1/ / 2, 1, 2,3

( )0 otherwise

iV x iG

⎧ ≤ Δ == ⎨⎩

x (2.26)

where Δ is the filter length and V is the volume defined by the filter length. The Gaussian

filter is given by

3

2 22 2( ) expG λ λ

π⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟Δ Δ⎝ ⎠ ⎝ ⎠

x x (2.27)

where λ is a constant commonly taken to be 6.

For variable density ρ , a mass-weighted Favre filtering is introduced according to

( ), ( ) ( , )V

t G t dρφ ρ φ= −∫x x y y y . (2.28)

Then, an instantaneous flow variable φ is decomposed into "φ φ φ= + where "φ is the

subgrid Favre-filtered component.

Note that the LES filtering has different properties than the RANS averaging, for instance,

' and 0φ φ φ≠ ≠ , (2.29)

and " and 0φ φ φ≠ ≠ . (2.30)

Besides, the derivation of balance equations for the filtered variables φ or φ requires the

exchange of filter and derivative operators. This exchange is valid only under restrictive

assumptions. For instance, the filtering operation only commutes with differentiation for a

uniform stationary grid (Ghosal and Moin, 1995). In other words,

17

x xφ φ∂ ∂=∂ ∂

(2.31)

holds only for a uniform stationary grid. In general, the uncertainties due to this operator

exchange are neglected and their effects are assumed to be incorporated in the subgrid scale

models.

Applying the filtering definition to the instantaneous Navier-Stokes equations for reactive

flows with sprays gives the filtered balance equations as follows.

( )j s

j

ut x

ρρ ρ∂∂

+ =∂ ∂

(2.32)

( ) ( )i j si

ij ij ij i j

u uu p Ft x x x

ρρ σ ρτ∂∂ ∂ ∂

+ = − + − +∂ ∂ ∂ ∂

(2.33)

( ) ( ) ( )

_______

, 1 1,j l

l j l c sl j l u Y l l l

j j

Y u YD Y l N

t x x

ρ ρρτ ω ω δ

∂ ∂ ∂ ⎛ ⎞+ = − + + =⎜ ⎟∂ ∂ ∂ ⎝ ⎠ (2.34)

( ) ( ), ,

1

j s

Ns j sj u h l j l s l

lj j j j j

c siij

j

h u h p p Tu D Y ht x t x x x x

u Q Qx

ρ ρλ ρτ ρ

σ

=

∂ ∂ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂+ = + + − − +⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ⎝ ⎠⎝ ⎠

∂+ +

∂

∑ (2.35)

where ijσ is the viscous stress term and is approximated as

2( )3

ji kij ij ij

j i k

uu ux x x

σ σ μ μ δ∂∂ ∂

≅ = + −∂ ∂ ∂

. (2.36)

18

where μ is the effective viscosity. siF is the filtered rate of momentum gain per unit volume

due to spray and requires approximation. ijτ ,j lu Yτ and

j su hτ are the sub-grid scale stress

tensor and sub-grid scale scalar fluxes, and are defined as follows

( )ij i j i ju u u uτ = − , (2.37)

( )j lu Y j l j lu Y u Yτ = − , (2.38)

and ( )j su h j s j su h u hτ = − . (2.39)

The filtering operation gives rise to a number of sub-grid scale terms that are unclosed. One

can simply ignore the sub-grid scale terms. For instance, one can ignore the sub-grid stress

tensor but rely on appropriate numerics to introduce the additional dissipation to account for

the unresolved field. But this approach is very difficult to analyze and therefore it is difficult

to choose the appropriate numerical scheme (Ghosal, 1998). More appropriate approach is to

directly model the sub-grid stress tensor in the momentum equation.

The filtered laminar diffusion fluxes for species and enthalpy may be either neglected or

modeled through a simple gradient assumption and uniform diffusivity as

____________

_______

, , ,l l l

l j l l j l jj j j

Y Y YD Y D D Dx x x

ρ ρ ρ∂ ∂ ∂= − ≈ − ≈ −

∂ ∂ ∂ (2.40)

j j

T Tx x

λ λ∂ ∂=

∂ ∂ (2.41)

19

Other filtered non-linear terms also need models for the closures. For instance, unresolved

Reynolds stresses ( )i j i ju u u u− require subgrid scale turbulence models, and unresolved

fluxes ( )j l j lu Y u Y− and ( )j s j su h u h− require subgrid scale scalar models. The following

will present various models that have been proposed to close these unresolved terms.

The sub-grid scale stress models can be classified into two major categories: constant

coefficient models and dynamic models. The constant coefficient models generally require a

priori specification of the model coefficient based on flow conditions and/or grid resolution.

Commonly-used constant coefficients models include the Smagorinsky model (Smagorinsky,

1963), the scale similarity (Bardina et al., 1980; Bardina et al., 1983), the mixed model

(Bardina et al., 1983), and the gradient model (Yeo, 1987). The dynamic models calculate

the model coefficient based on the resolved field instead of using a constant value. Examples

of this type include dynamic Smagorinsky model (Germano et al., 1991) and one-equation

non-viscosity structure model (Pomraning and Rutland, 2002).

Constant coefficient models

The Smagorinsky model assumes that the anisotropic part of the sub-grid stress tensor is a

scalar multiple of the filtered rate of strain tensor. The sub-grid scale stress tensor is then

given by

13ij ij ll t ijv Sτ δ τ− = − (2.42)

where tv is the eddy viscosity and defined as

20

22t sv C S= Δ , (2.43)

where Δ is the filter width, and ijS is the filtered rate of strain tensor

12

jiij

j i

uuSx x

⎛ ⎞∂∂= +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

, (2.44)

and S is the magnitude of the filtered rate of strain tensor given by

( )1/ 22 ij ijS S S= . (2.45)

The parameter sC is a user-specified coefficient that may depend on flow configuration and

grid resolution. The Smagorinsky model has the following drawbacks. First, the assumption

based on a molecular transport analogy cannot be justified. In fact, the real sub-grid stresses

do not align with the filtered rate of strain and the correlations between them are low (Liu et

al., 1994). Second, it is very difficult to specify an optimal value for the constant sC for

complex turbulent flows under different simulation conditions. Third, the model is purely

dissipative, i.e., energy only flows from the resolved scale to the sub-grid scale. This

characteristic may be helpful to numerical stability but neglect an important physics – reverse

energy cascade or backscatter in which the energy is transferred from the sub-grid to the

resolved scales (Piomelli et al., 1991). A good model should be able to account for the

backscatter of energy.

Bardina (1980 and 1983) proposed the scale similarity model that addresses the backscatter

of energy flows. The sub-grid scale stresses can be approximated from the resolved field

(Yeo, 1987) based on the assumption that the sub-grid stresses are similar to the smallest

21

resolved stresses. The scale-similarity model is shown to correlates reasonably well with the

sub-grid scale stresses. However, it does not ensure an overall positive transfer of energy to

the sub-grid scales (Bardina et al., 1983). It is often coupled to the Smagorinsky model to

derive a mixed model to address the insufficient energy dissipation of the scale similarity

model (Bardina et al., 1983). As with the Smagorinsky model, the mixed model also needs

to specify the model constant sC .

The gradient model (Bedford and Yeo, 1993) relates the sub-grid stress tensor to the gradient

of velocity. The gradient model correlates well with the actual sub-grid stress tensor due to

the reason that this model is the first term in a series expansion for the exact sub-grid stress

tensor (Pomraning, 2000). However, this model may result in a negative viscosity (Leonard,

1997) which can cause the model to become unstable.

Dynamic models

Since the model coefficient sC in Eq. (2.43) usually depends on the flow and grid resolution,

it will be more appropriate to directly calculate it instead of using a universal constant. As a

result, dynamic subgrid models were developed based on the work by Germano et al. (1991).

This dynamic approach is based on an algebraic identity between the subgrid scale stresses at

two different filtered levels and the resolved turbulent stresses. Two filtering operations are

performed to formulate a dynamic model. In addition to the ‘grid’ filtering operation,

indicated by an overbar , that leads to the filtered LES governing equations, a second

22

filtering called ‘test’ filtering (indicated by an arc ) is performed on the resolved field. The

‘test’ filtering is defined in a similar manner as the ‘grid’ filtering operation as

( ) ( ) ( )TVG dφ φ= −∫x x y y y , (2.46)

where, for the ‘box’ filter, the function TG becomes

1/ / 2, 1, 2,3( )0 otherwise.

iT

V x iG⎧ ≤ Δ =⎪= ⎨⎪⎩

x (2.47)

V is the ‘test’ filter volume. Δ is the ‘test’ level filter length which is usually twice the

‘grid’ level filter length Δ (Germano et al., 1991). Based on this definition, a stress tensor at

the ‘test’ level can be defined by

( )ij i j i jT u u u u= − (2.48)

Applying the ‘test’ filtering on the subgrid scale stress ijτ gives

( )ij i j i ju u u uτ = − . (2.49)

Further, the Leonard stress term ijL is defined as

( )ij i j i jL u u u u= − . (2.50)

The elements of ijL can be calculated from the resolved velocity field. This is accomplished

by invoking the ‘test’ filtering on the resolved field with a filter size typically twice the grid

filter. Then, the ‘grid’ level stress tensor and the ‘test’ level stress tensor are related by the

Germano identity (Germano et al., 1991)

ij ij ijL T τ= − . (2.51)

23

This identity can be used to derive a dynamic Smagorinsky model that calculates the

coefficient sC in Eq. (2.43) as a function of time and space (Germano et al., 1991; Moin et

al., 1991; Lilly, 1992). Assuming the same functional form to parametrize both ijT and ijτ ,

the ‘test’ level stress tensor can be approximated as

21 23ij ij ll s ijT T C S Sδ− = − Δ . (2.52)

Then, the Germano identity yields the following form

1 23ij ij kk s ijL L C Mδ− = − , (2.53)

where

2 2ij ij ijM S S S S= Δ −Δ . (2.54)

Germano et al. (1991) multiplies ijS on both sides of Eq. (2.53) to obtain a dynamic

coefficient as

12

ij ijs

mn mn

L SC

M S⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

. (2.55)

Lilly (1992) applies a least square approach to obtain an improved coefficient given by

12

ij ijs

mn mn

L MC

M M⎛ ⎞

= ⎜ ⎟⎝ ⎠

. (2.56)

The dynamic coefficients sC may be positive or negative. A negative coefficient can lead to

numerical instability that is usually addressed with additional averaging either in

homogeneous spatial directions (Germano et al., 1991) or in time (Meneveau et al., 1996).

For numerical stability, Piomelli and Liu (1995) arbitrarily set the negative coefficients to

24

zero thus eliminating all of the effects of ‘backscatter’. The results were more accurate and

more computationally efficient than a planar averaged model (Piomelli and Liu, 1995). For

complex flows with no homogeneous directions, Meneveau et al. (1996) proposed a

Lagrangian dynamic model that is derived by minimizing the dynamic Smagorinsky model

error along flow pathlines in a Lagrangian frame of reference.

To address the numerical instability associated with the reverse energy flow from the sub-

grid to the resolved scales, a transport equation for the sub-grid scale kinetic energy can be

used to enforce a budget on the energy flow between the resolved and the sub-grid scales.

The sub-grid kinetic energy k is defined as

( )12 i i i ik u u u u= − (2.57)

This quantity cannot directly be calculated from the resolved velocity field and must be

modeled. A transport equation for k can be derived by following the filtered momentum

equation as (Yoshizawa and Horiuti, 1985; Menon et al., 1996; Warsi, 2006)

j tij ij

j j k j

u kk k St x x x

ν τ εσ⎛ ⎞∂∂ ∂ ∂

+ = − −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ (2.58)

where the terms on the right-hand side of Eq. (2.58) represent diffusion, production, and

dissipation of the subgrid kinetic energy k , respectively. The sub-grid eddy viscosity tν and

the sub-grid kinetic energy dissipation rate ε are modeled based on the sub-grid kinetic

energy and the grid size as

1/ 2t kC kν = Δ (2.59)

25

3/ 2kCεε =Δ

(2.60)

1 3VΔ = (2.61)

where kC ,Cε and kσ are model constants with the values of 0.05, 1.0 and 1.0, respectively

(Yoshizawa and Horiuti, 1985), Δ is the filter width, and V is the volume of a computational

cell. This method is believed to be able to account for the non-equilibrium effect between

production and dissipation at the sub-grid level to improve the model accuracy for high

Reynolds number flows using relatively coarse grids (Menon et al., 1996). The reason is that

sub-grid information and a kinetic energy budget are available for the formulation of sub-grid

scale models, resulting in improved modeling of the effects of the sub-grid on the resolved

scales. However, Menon et al. (1996) reported that the current model for the dissipation rate

ε did not perform well. Thus, it is important to improve its modeling when the transport

equation is solved for k .

Using this transport equation, a non-viscosity model, called the dynamic structure model

(Pomraning and Rutland, 2002), was developed to estimate the stress tensor directly with

2ij ij

kk

k LL

τ = (2.62)

where the Leonard stress tensor ijL is rescaled by both its trace kkL and the trace kkτ of

ijτ ( )2kk kτ ≡ . This model was shown to provide a better scaling between the real stresses

and the modeled terms than the viscosity-based models such as dynamic Smagorinsky

models. The resolved and subgrid kinetic energies were shown to agree well with a direct

numerical simulation of decaying isotropic turbulence (Pomraning and Rutland, 2002).

26

Meanwhile, similar modeling approach can be extended to scalar transport modeling

(Chumakov and Rutland, 2004).

2.3 Spray Modeling

Engine sprays are typically composed of a very large number of droplets, and each of them

has its own properties and is subject to very complex interactions with flows. Due to limited

computational resources, it is not practical to account for each individual droplet in the

simulation. Two major strategies that have been formulated to describe sprays in CFD

simulations are Eulerian and Lagrangian formulations. The Eulerian formulation treats both

sprays and the gas phase as continuous fields (Spalding, 1981; Drew, 1983). For poly-

dispersed droplets, a number of separate fields can be used to represent different classes of

droplet sizes. The spray formulations proposed by Sinnamon et al. (1980), Hallmann et al.

(1995), Divis and Macek (2004), and Divis (2005) are examples of the Eulerian spray

formulation. The Eulerian formulation works best in situations where the resolution scales

are much larger than the average droplet spacing (Sirignano, 1986). Moreover, this

formulation suffers from numerical diffusion of particle density, especially in the vicinity of

the injector on coarse grids (Dukowicz, 1980; Loth, 2000).

Another strategy of spray modeling is the Lagrangian formulation based on a discrete-

particle model introduced by Dukowicz (1980), also known as the stochastic particle model

or discrete droplet model. The spray is represented by collective computational parcels with

each parcel consisting of a number of droplets that are assumed to have identical properties

27

such as position, radius, density, velocity, and temperature. Each parcel is tracked in the

Lagrangian frame of reference. The coupling between the spray and gas phases are achieved

by exchanging source terms for mass, momentum, energy, and turbulence. Various sub-

models are developed to account for the effects of droplet breakup, collision, turbulent

dispersion, vaporization, and wall interaction. The properties of the representative parcels

are randomly chosen from empirical or theoretical distribution functions. An adequate

statistical representation of realistic sprays may be obtained when a sufficiently large number

of parcels are used (Watkins, 1987). In addition, the Lagrangian formulation assumes large

void fraction and thus is suitable for dilute sprays but has drawbacks in modeling dense

sprays (Amsden et al., 1989). To address the dense spray near the injector exit, extended

Lagrangian formulations (Stalsberg-Zarling et al., 2004; Gavaises and Tonini, 2006) or the

Eulerian dense spray coupled with the Lagrangian dilute spray (Wang and Peters, 1997;

Blokkeel et al., 2003; Lebas et al., 2005) is also developed. The Lagrangian approach is the

most common spray description in engine simulation and is also used in this study. The

following will review various spray sub-models used to simulate the engine spray processes.

Breakup models

As the spray enters the domain and interacts with the surrounding gas, it will experience the

so-called atomization process in which the spray breaks up into small droplets. Atomization

can be divided into two main processes: primary breakup and secondary breakup. The

primary breakup occurs near the injector at high Weber numbers and is determined by not

only the gas-liquid interaction but also the phenomena inside the nozzle such as cavitation

28

and turbulence. The secondary breakup occurs downstream of the spray and is mainly due to

aerodynamic effects.

Primary breakup

The purpose of a primary breakup model is to determine the starting conditions of the drops

that are created at the nozzle exit.

DUinj

blob

Secondary break-upNozzle hole

(a) Schematic view of blobs

y

z

x

Uinjφ/2

φ

ψ

(b) Spray cone angle in spherical coordinate system Figure 2.2. Blob-method of primary breakup.

Blob method

A simple but popular model for defining the starting conditions of a spray is the so-called

“blob method” (Reitz, 1987; Reitz and Diwakar, 1987). This method assumes that the initial

fuel jet breakup and drop breakup near the nozzle exit are indistinguishable processes, and

that a detailed simulation can be replaced by the injection of big spherical drops with a

uniform size. These drops are then subject to secondary breakup as shown in Figure 2.2 (a).

The size of the blobs is equal to the nozzle diameter. The number of drops injected per unit

time is determined from the mass flow rate. The velocity components of the blobs are

determined using the spray cone angleφ . The direction of the drop velocity inside the 3D

29

spray cone is randomly chosen by using two random numbers to predict the azimuthal angle

ϕ and the polar angle ψ in the spherical coordinate system as shown in Figure 2.2 (b).

In real engine sprays, particularly high-pressure diesel injection, cavitations often occur

inside the nozzle affecting the actual flow area and injection velocity. The above blob

method can be improved by calculating effective velocity and exit area resulting from

cavitation (Kong et al., 1999; Kuensberg et al., 1999). Applying momentum and mass

conservations between point 1 and point 2, as shown in Figure 2.3, gives the effective

injection velocity by

( )2injhole

eff vapinj l hole c

mAu p pm A Cρ

= − + (2.63)

where vapp is the vapor pressure and cC is the coefficient determined according to (Nurick,

1976). The effective flow area effA and blob diameter effD are calculated as

1/ 24

and Dinj effeff eff

eff l

m AA

u ρ π⎛ ⎞

= = ⎜ ⎟⎝ ⎠

. (2.64)

AholeAeffAhole·Cc1 2

ueffuvena

Inside injector

Nozzle exit

cavitation Figure 2.3. 1-D approximation of internal nozzle flow with cavitation.

30

The blob method is simple in terms of the conceptual understanding and code

implementation. However, it does not provide detailed information about the influence of

the phenomena inside the nozzle. This method also does not consider the promotion of

primary breakup by turbulence and by implosions of cavitation bubbles inside the liquid jet.

Turbulence-induced breakup

This model is based on the phenomenological model developed by Huh and Gosman (1991)

for predicting the primary spray angle of solid-cone diesel sprays. It assumes that the

turbulence force within the liquid emerging from the nozzle can create initial surface

perturbations that grow exponentially due to aerodynamic forces to form new droplets. The

turbulent length scale determines the wavelength of the most unstable surface wave. The

turbulent kinetic energy at the nozzle exit is determined based on mass, momentum, and

energy balances. Huh and Gosman (1991) also assumed that the spray expands with a radial

velocity. The spray angle can be determined by combining the radial velocity and axial

velocity. The direction of the resulting velocity of the primary blob inside the spray cone is

randomly chosen as described for the blob method. The model also assumes the injection of

spherical blobs with the diameter equal to the nozzle hole diameter. The final velocity

components are determined using the average turbulent kinetic energy and its dissipation rate

at the nozzle exit.

The model determines the jet breakup based on a characteristic length scale AL and time

scale Aτ . The length scale AL is assumed to be proportional to the turbulence length scale

31

and the time scale Aτ is a linear combination of the turbulence time scale and the wave

growth time scale. The wave growth time scale is determined by applying the Kelvin-

Helmholtz instability theory to an infinite plane. The turbulence length scale and time scale

are determined from turbulence modeling. The breakup rate of the primary blob is

proportional to the atomization length and time scale by

( ) 1A

Adrop

Ld d kdt τ

= (2.65)

where 1 0.05k = and dropd is the diameter of the new drops. Both AL and Aτ depend on time

since the internal turbulence of parent drops decays with time as they travel downstream.

Cavitation-induced breakup

The caviation-induced model was developed by Arcoumanis et al. (1997) for solid-cone

diesel sprays to model the primary breakup by accounting for cavitation, turbulence, and

aerodynamic effects. This model assumes that the liquid jet emerging from the nozzle exit

breaks up due to the collapsing cavitation bubbles. A 1-D submodel was used to estimate the

nozzle exit data such as the effective hole area effA , injection velocity injU , and turbulent

kinetic energy k and its dissipation rate in order to link the nozzle flow to the spray

characteristics. The initial droplet diameter is assumed equal to the effective hole diameter.

The first breakup of these blobs is modeled using the Kelvin-Helmholtz mechanism for

aerodynamic-induced breakup, the model of Huh and Gosman (1991) for turbulence-induced

breakup, and a phenomenological model for cavitation-induced breakup. This model also

assumes that the cavitation bubbles either burst on the jet surface or collapse before reaching

32

it due to the smaller pressure compared to the ambient pressure. The minimum of the

characteristic time scales between the two cases is used to determine breakup (Arcoumanis et

al., 1997).

In the case of bubble collapse, the cavitation bubbles are lumped into a single big “effective”

bubble having the same area as the total area of all the actual bubbles. The radius of this

artificial bubble is used to estimate the collapse time collτ from the Rayleigh theory of bubble

dynamics (Brennen, 1995). In the case of bubble bursting, the “effective” bubble is placed in

the center of the liquid and then moves radially to the liquid surface with a turbulent velocity

2 / 3turbu k= , resulting in a burst time burstτ . The characteristic atomization time scale is

taken as ( )min ,A coll burstτ τ τ= . The atomization length scale is given by a correlation based

on the nozzle hole size and cavitation bubble size. The breakup force acting on the jet

surface at the time of collapsing or bursting of a cavitation bubble can then be estimated from

dimensional analysis. The breakup force is used together with the surface tension force to

determine the primary breakup. The exact size of the new droplets is sampled from a chosen

distribution function.

Cavitation and turbulence-induced breakup

Nishimura and Assanis (2000) developed a cavitation and turbulence-induced primary

breakup model for solid-cone diesel sprays that considers cavitation bubble collapse energy,

turbulent kinetic energy, and aerodynamic forces. The model injects discrete cylindrical

ligaments which have a diameter D and a volume equal to that of a sphere with the same

33

diameter as seen in Figure 2.4. The bubbles collapse and energy is released, resulting in an

increase of turbulent kinetic energy colk . Assuming isotropic turbulence, the turbulent

velocity inside the liquid cylinder can be determined as

( )23

col nozzturb

k ku

+= (2.66)

where nozzk is the turbulent kinetic energy at the nozzle exit. This turbulent velocity is used

to determine the deformation force turbF . The liquid cylinder breaks up if the sum of turbF

and the aerodynamic drag force no longer competes with the surface tension force.

Figure 2.4. Primary breakup model of Nishimura and Assanis (2000).

Other authors also developed similar models for solid-cone diesel sprays based on different

mechanisms. For instance, the model by Von Berg (2002) releases droplets from a coherent

core. The sizes and velocity components of the droplets are calculated based on local

turbulent scales and breakup rates on the core surface. Baumgarten et al. (2002) has

developed a detailed cavitation and turbulence-induced model that is based on energy and

34

force balances and predicts the initial spray conditions needed to simulate further secondary

breakup (Baumgarten, 2006).

Figure 2.5. (a) Inwardly opening pressure-swirl atomizer, (b) outwardly opening pressure-

swirl atomizer (Baumgarten, 2006).

Sheet atomization model for hollow-cone sprays

In direct-injection spark ignition engines, hollow-cone sprays are usually generated to

maximize dispersion of the liquid phase at moderate injection pressures. Hollow-cone sprays

are typically characterized by small droplet diameters, effective fuel-air mixing, reduced

penetration, and thus high atomization efficiency. A model developed by Schmidt et al.

(1999), usually referred to as the Linearized Instability Sheet Atomization (LISA), is often

used to describe the primary breakup of hollow-cone sprays. In this model, a zero-

dimensional approach is used to represent the internal injector flow and to determine the

velocity at the nozzle hole exit. In the region near the exit, the transition from the injector

flow to the fully developed spray is modeled by a three-step mechanism: film formation,

sheet breakup, and disintegration into droplets (Senecal et al., 1999), as can be seen in

Figure 2.5. More information about this model and the modeling of hollow-cone sprays can

be found in (Schmidt et al., 1999; Senecal et al., 1999).

35

Secondary breakup

Secondary breakup is the breakup of the current existing droplets into smaller ones due to the

aerodynamic forces that are induced by the relative velocity relu between the droplet and

surrounding gas. Secondary breakup is believed to be controlled by a balance of two forces

acting on the droplet that lead to opposite effects. The aerodynamic forces results in an

unstable growth of waves on the droplet surface that leads to the disintegration. The surface

tension force on the other hand tends to maintain the spherical droplet and counteracts the

deformation force. The ratio of aerodynamic forces and surface tension force is a non-

dimensional parameter, the gas phase Weber number gWe

2

g rel dg

u dWe

ρσ

= . (2.67)

Depending on the Weber number, different regimes and mechanisms of droplet breakup can

exist. The following several paragraphs present models used to simulate the breakup process

in sprays.

Phenomenological models

Arcoumanis et al. (1997) divided droplet breakup into seven different regimes based on the

Weber number. Using phenomenological relationships the breakup times is estimated as

d lbr break

rel g

dtu

ρτρ

= (2.68)

where the dimensionless breakup time breakτ is given in Table 2.1. If the breakup time is

greater than the life time of the droplet, the breakup occurs. The product droplet sizes are

36

sampled from distribution functions. The Sauter mean diameter (SMD) of the droplets is

estimated using phenomenological relations based on the Weber number, drop diameter, drop

viscosity, relative velocity, and densities of the drop and gas. In the case of stripping

breakup, small product droplets are stripped from the parent droplets, and their mass is

subtracted from their parent drops.

Table 2.1. Breakup regimes and breakup times of droplets (Arcoumanis et al., 1997).

Breakup regime Breakup time breakτ Weber number

Vibrational 0.5

3 20.25 6.25 lbreak

l ld dμστ π

ρ ρ

−⎡ ⎤

= −⎢ ⎥⎣ ⎦

12gWe ≤

Bag ( ) 0.256 12break gWeτ

−= − 12 18gWe< ≤

Bag-and-steamen ( )0.252.45 12break gWeτ = − 18 45gWe< ≤

Chaotic ( ) 0.2514.1 12break gWeτ

−= − 45 100gWe< ≤

Sheet stripping ( ) 0.2514.1 12break gWeτ

−= − 100 350gWe< ≤

Wave crest stripping ( )0.250.766 12break gWeτ = − 350 1000gWe< ≤

Catastrophic ( )0.250.766 12break gWeτ = − 1000 2760gWe< ≤

5.5 2760 gWe<

Taylor Analogy Breakup (TAB) Model

The TAB model developed by O’Rourke and Amsden (1987) is based on an analogy between

an oscillating and distorting drop that penetrates through surrounding gas with a relative

velocity relu and a forced oscillating spring-mass system. The aerodynamic drag acts as the

external force, the surface tension as the restoring force, the liquid viscosity as the damping

37

force. Using this analogy, the equation for the acceleration of the droplet distortion

parameter y is

2

2 3 2g rell F

d kl l b l

uCy C y C yr r C r

ρμ σρ ρ ρ

+ + = (2.69)

where y is the normalized displacement of the droplet’s equator from its equilibrium

position. dC , kC , bC , and FC are model constants. r is the droplet radius in spherical shape.

gρ and lρ are the gas and liquid densities, and σ is the surface tension. The size of the

product droplets is estimated using an energy balance. For each breakup event, the radius of

the product drops is chosen randomly from a -squaredχ distribution. The number of the

product drops can then be determined using mass conservation.

The TAB model is one of the original drop breakup models and is often used to predict

gasoline drop breakup. This model can underpredict the droplet size of diesel sprays (Liu

and Reitz, 1993; Tanner, 1997) and underestimate the spray penetration (Park et al., 2002).

An enhanced TAB model (ETAB) was proposed by Tanner (1997) that accounts for the

initial oscillation in addition to the modified product drop sizes. The deformation velocity is

also modified to increase the lifetime of the blobs and to allow a more realistic representation

of the dense spray near the nozzle exit (Baumgarten, 2006). Ibrahim (1993) also extended

the TAB model by considering the extensional flow that causes the drop to become an oblate

spheroid. A different breakup criterion was formulated and the product droplet size was

estimated using the same method as in the TAB model.

38

Kelvin Helmholtz (KH) Breakup Model

The KH model is based on a first order linear analysis of a Kelvin-Helmholtz instability

growing on the surface of a cylindrical liquid jet that is penetrating into a stationary gas.

These surface waves grow due to the aerodynamic force resulting from the relative velocity

between the liquid and gas as shown in Figure 2.6. It is assumed that the wave with the

highest growth rate is the most unstable surface wave and will cause breakup to form new

droplets. The growth rate Ω of this wave can be determined from the numerical solution to a

dispersion function of a perturbation (Reitz and Bracco, 1986) according to

( )( )

50.5 6 1.530

0.6

0.34 0.38 4 123 1 1 1.4

gl

break

Wery Z T

ρσ π

+⎛ ⎞⎡ ⎤ ⎛ ⎞Ω = =⎜ ⎟⎜ ⎟⎢ ⎥ + +⎝ ⎠⎣ ⎦ ⎝ ⎠, (2.70)

and the corresponding wavelength Λ ,

( )( )

( )

0.5 0.7

0.61.670

1 0.45 1 0.4 9.02

1 0.87 g

Z Tr We

+ +Λ=

+, (2.71)

where We, Z and T are the Weber, Ohnesorg and Taylor number, respectively, and are

defined as

2

0

0

, ,g rel lg g

l

r uWe Z T Z We

rρ μσ ρ σ

= = = . (2.72)

0r is the radius of the undisturbed jet. The radius of the new droplets is assumed to be

proportional to the wavelength Λ as

0newr B= Λ , (2.73)

where 0 0.61B = is a constant. In contrast to the complete breakup of the parent drop in the

TAB model, the parent drop continuously loses mass while penetrating into the gas. The

39

radius shrinks at a rate that depends on the difference between the current droplet radius r

and the child droplet radius newr and on the characteristic time

1, 3.788 newbu

bu

r rdr rBdt

ττ−

= − =ΛΩ

. (2.74)

where buτ is the breakup time scale and 1B is an adjustable constant to account for the

influence of the internal nozzle flow on the spray breakup.

Figure 2.6. Schematic illustration of drop breakup mechanisms (a) KH-type; (b) RT-type

(Kong et al., 1999).

The KH model creates a group of drops that demonstrate a bimodal distribution of drop sizes

with a small number of big parent drops and an increasing number of small child droplets.

Although stripping breakup is one of the most important breakup mechanisms for high-

pressure injection, experiments have shown (Hwang et al., 1996) that the sudden

disintegration of a complete drop into droplets with diameter much bigger than the KH-child

droplets is also important. For this reason, the KH model is usually combined with the

Rayleigh-Taylor model that will be described next.

40

Rayleigh Taylor (RT) Breakup Model

The RT model is based on the work of Taylor, as cited by Batchelor (1963), about the

instability of the interface between two fluids of different densities with acceleration being

normal to this interface. For drop and gas moving with a relative velocity relu , the drop

deceleration due to drag forces can be viewed as an acceleration of the drop in the opposite

direction (airflow direction). The Taylor’s theory indicates that the interface can be unstable

if the acceleration is directed into the gas. Thus, unstable waves can grow on the back side of

the drop, as seen in Figure 2.6 (b). From the aerodynamic drag force, the acceleration of the

drop can be simplified to

23

8g rel

Dl

ua c

rρρ

= . (2.75)

Assuming linearized disturbance growth rates and negligible liquid viscosity, the growth rate

Ω and the corresponding wavelength Λ of the fastest growing wave are:

1/ 42

3 3laa ρ

σ⎡ ⎤Ω = ⎢ ⎥⎣ ⎦

, (2.76)

and

33 2

l

Caσπρ

Λ = . (2.77)

The breakup occurs only if dropdΛ < . The breakup time 1but −= Ω is the reciprocal of the

frequency of the fastest growing wave. At but t= the drop breaks up completely into smaller

drops with the radius newd = Λ and the number of new drops is obtained based on the mass

conservation (Patterson and Reitz, 1998). C3 is an adjustable constant to control the

41

effective wavelength by accounting for unknown effects such as the initial disturbance at the

nozzle exit.

Drag-deceleration (RT model) and shear flow (KH model) induced instability phenomena

have been observed simultaneously in the droplet breakup process (Hwang et al., 1996).

Hence, the RT model is always used in combination with a second breakup model, usually

the KH model. In addition, combined models have been widely used because no single

breakup model can describe all relevant classes of breakup processes and regimes accurately.

The combined KH-RT breakup model uses the KH model near the nozzle exit and uses the

RT model beyond the breakup length bL to avoid a too fast reduction of the drop size near

the nozzle exit (Kong et al., 1999), as shown in Figure 2.7.

DUinj

Breakup length: LbNozzle hole

KH break-up

RT break-up

Ø/2

Figure 2.7. Combined blob-KH/RT model (Kong et al., 1999).

Collision and coalescence

Droplet collision plays an important role in influencing spray dynamics, particularly in the

dense spray region where the probability of collision is high. The result of a collision event

depends on the impact energy, the ratio of droplet sizes, and ambient conditions including

42

gas density, gas viscosity, and the fuel-air ratio (Baumgarten, 2006). Collision can result in

coalescence, pure reflection, or breakup. As a result of collision, the droplet velocity,

trajectory, and size will be changed.

Figure 2.8. Regimes and mechanisms of droplet collision (Baumgarten, 2006).

The collision phenomenon can be characterized by using four dimensionless parameters: the

Reynolds number, the Weber number, the drop diameter ratio, and the impact parameter.

The possible outcomes of a collision event include bouncing, coalescence, reflective

separation, stretching separation, and shattering, as shown in Figure 2.8. Most of the

practical collision models do not consider all of the above phenomena due in part to the fact

that it is impossible to directly evaluate a collision model by comparison with available

experimental data. The standard collision model used in most spray simulations is the model

by O’Rourke and Bracco (1980), which considers only two outcomes: coalescence and

43

stretching separation (grazing collision). This model follows the approach of Brazier-Smith

et at. (1972) and uses an energy balance to predict whether the two colliding drops separate

again after coalescence to re-form the original drops or combine to form a larger drop.

Tennison et al. (1998) enhanced this model by taking into account reflective separation.

Georjon and Reitz (1999) proposed a drop-shattering collision model by extending the

stretching separation regime.

The O’Rourke model is often implemented by using the statistical approach for enhanced

efficiency. The implementation of the O’Rourke model using the statistical approach is

inherently grid dependent based on three factors. First, droplets can only collide within the

same computational cells. Second, the collision frequency depends on the size of grid cells.

Third, the implementation uses the magnitude of the relative velocities without considering

the orientation of the velocities. These factors have been shown to cause several artifacts that

were observed in engine spray simulations (Schmidt and Rutland, 2000; Hieber, 2001;

Nordin, 2001; Are et al., 2005). Advanced numerical schemes such as adaptive mesh

refinement can be used to alleviate such grid dependence as will be discussed later.

Vaporization models

The vaporization of liquid fuel influences ignition, combustion, and formation of pollutants.

The vaporization process involves heat transfer and mass transfer that will affect temperature,

velocity, and vapor concentration in the gas phase. The standard modeling approach is to use

a single component vaporization model, e.g., tetradecane for diesel fuel and iso-octane for

gasoline. The rate of the drop temperature change is described as

44

( )3

2 24 4 43

dd l d d d

dTr drc r Q r L Tdt dt

π ρ π ρ π= + (2.78)

where lc is the specific heat of the liquid, r is the radius of the drop, ( )dL T is the latent heat

of vaporization, and dQ is the rate of heat transfer to drop surface per unit area by conduction

and is obtained by using the Ranz-Marshall correlation (Faeth, 1977)

( ) ( )3/ 2

1

2

ˆˆ ˆ, ˆ2g d

d g d g

T T K TQ K T Nu K Tr T K−

= =+

. (2.79)

The film temperature T̂ , the thermal conductivity ( )ˆgK T , and the Nusselt number dNu are

related to other fundamental properties and the Spalding’s mass transfer number of the drop

(Faeth, 1977). 1K and 2K are constants.

More sophisticated vaporization models consider the effect of multicomponents in the fuel.

These models include “discrete component” and “continuous component” approaches

(Sazhin, 2006). The first approach uses distinct full components to compose the drop

properties (Torres et al., 2003). The second approach uses a continuous thermodynamics

approach to describe the multicomponent effects of fuels (Tamin and Hallett, 1995).

Multicomponent vaporization models have the potential to describe the vaporization process

more accurately under a low temperature environment in which light components vaporize

earlier and influence ignition more significantly (Zhang and Kong, 2009).

45

Turbulence dispersion models

Turbulence affects not only the gas phase flow but also the motion of droplets. A portion of

turbulent kinetic energy is used to disperse the droplets and the droplet-turbulence

interactions need be considered in spray modeling. The behavior of droplets in the flow field

can be characterized by a non-dimensional parameter called the Stokes number St

r

e

St ττ

= (2.80)

where rτ and eτ are the aerodynamic response time and the eddy life time, respectively. The

aerodynamic response time rτ indicates the responsiveness of a droplet to a change in gas

velocity and the eddy life time eτ is an eddy breakup time equal to /k ε . For small Stokes

number, the droplets react to the flow very quickly and will follow the flow field very well.

For large Stokes number, the droplets react to the flow change slowly and the droplets can

barely follow the flow field.

In the Lagrangian-Eulerian approach, a droplet generally evolves on the aerodynamic

response time

r

ddt τ

−= g dd u uu (2.81)

where gu is the instantaneous gas velocity. A droplet interacts with a wide range of

turbulence length and time scales. Pai and Subramaniam (2006) found that a model evolving

droplets over the response time using a linear drag law will predict an anomalous increase in

the liquid phase turbulent kinetic energy for decaying turbulent flow laden with droplets. Pai

and Subramaniam (2006) proposed that the fluctuating droplet velocity relaxes to the local

46

modeled fluctuating gas phase velocity on a multiscale interaction time scale. Numerical

tests on a zero-gravity, constant-density, decaying homogeneous turbulent flow laden with

sub-Kolmogorov-size droplets show that the improved model predicts correct trends of the

turbulent kinetic energy for both phases.

Wall interaction models

Spray-wall interactions can influence mixture distribution, combustion, and emissions. The

possible outcomes include stick, rebound with or without breakup, wall jet, spread, splashing,

and wall film formation (Kong, 2007). Many factors can influence the outcomes of the

spray-wall interactions, such as the incident drop velocity, incidence angle, liquid density,

surface tension, and wall temperature and wettability. These factors can be combined to

form several important dimensionless parameters: Weber, Reynolds, and Ohnesorge numbers.

The Weber number is the most important parameter and its definition for the drop

impingement represents the ratio of the droplet’s normal inertia to its surface tension as

2

l nl

u dWe ρσ

= (2.82)

where nu is the drop’s velocity component normal to the surface, lρ is the liquid density, σ

is the surface tension, and d is the drop diameter. Different impingement criteria can be

formulated based on the drop Weber number.

Naber and Reitz (1988) developed an impingement model that considers the stick, rebound

and spread regimes based on Weber number. This model was improved later by correcting

the normal drop velocity in the rebound regime. Bai and Gosman (1995) developed a model

47

that considers the splash regime. This model also combines the stick and spread regimes as

adhesion regime for a dry wall. Other detailed wall-impingement models have been

developed by O’Rourke and Amsden (2000), Lee and Ryou (2001), and Stanton and Rutland

(1996). In these models, the splash regime is treated differently than that by Bai and Gosman

(1995).

2.4 Combustion Modeling

The objective of the combustion modeling is to close the mean reaction rate lω in Eq. (2.10)

in the RANS approach or the filtered reaction rate clω in (2.34) in the LES approach. Since

the reaction rate is usually highly nonlinear, the mean or filtered reaction rate cannot be

simply expressed as a function of the mean quantities of the flow field and the species. Thus,

appropriate models need to be developed to address unclosed terms resulting from RANS

averaging or LES filtering. The model development is strongly dependent on the relative

magnitude of characteristic chemical time scales cτ and turbulent time scales tτ since the

interactions of chemistry and turbulence occur in a wide range of time scales in an engine

combustion. The relative magnitude of the time scales can be divided into cases: slow

chemistry (chemical time scales much larger than turbulent time scales), finite rate chemistry,

and fast chemistry (chemical time scales much smaller than Kolmogorov time scale ητ )

(Williams, 1985; Peters, 2000; Pope, 2000; Poinsot and Veynante, 2001; Fox, 2003). A

dimensionless parameter called the Damkohler number /a t cD τ τ= can be used to represent

48

the above relations. The Damkohler number along with other dimensionless parameters such

as Reynolds number can be used to define different combustion regimes.

Combustion requires that fuel and air be mixed at the molecular level. Depending on how

well the fuel-air mixture is prepared before the occurrence of combustion, engine

combustions can be divided into three categories: premixed combustion (i.e., conventional

gasoline engines), non-premixed combustion (i.e., conventional diesel engines), and partially

premixed combustion.

Closure models in RANS approach

For premixed flames, the simplest approach is to directly use the mean local values of density

and species to represent the mean Arrhenius reaction rates by neglecting the turbulence

effects. The approach is relevant only in the case of slow chemistry. If turbulence plays a

rate-limiting role, then the so-called eddy breakup model (Spalding, 1977; Peters, 2000) can

be used to represent the mean reaction rate as

( )1eCkθεω ρ θ θ= − (2.83)

where θ is a progress variable ( 0 : fresh gases and 1: burnt gasesθ θ= = ) and θ is its mean

and solved from a transport equation, k and ε are the turbulent kinetic energy and its

dissipation rate, and eC is a constant. This model is simple but useful in many applications.

However, it does not include effects of chemical kinetics and it tends to overestimate the

reaction rate, especially in highly strained regions. There are also other more complicated

49

models that require solving a number of transport equations such as the probability density

function (PDF) models (Pope, 2000; Fox, 2003).

For non-premixed flames, mixing of fuel and air is important and generally limits the

chemical reactions. Non-premixed flames also do not exhibit well-defined characteristic

scales. These factors make it more difficult to define combustion regimes (Poinsot and

Veynante, 2001; Fox, 2003). Since the overall reaction rate is limited by both the chemistry

and the molecular diffusion of species toward the flame front, the model needs to consider

both chemistry and turbulent effects. Two popular models of this type are the flamelet model

and the eddy dissipation concept model. In the flamelet model, the instantaneous reaction

rates for species can be expressed as a function of the scalar dissipation rate χ and the

mixture fraction z (Peters, 1984; Peters, 2000; Kong et al., 2007b) only

2

2

12

ll

Yz

ω ρχ ∂= −

∂ (2.84)

where lY is the mass fraction for species l . Flow influence is completely determined by the

scalar dissipation rate χ , which is defined as 2D C Cχ = ∇ ∇i where C is a conserved scalar

and D is the scalar diffusivity. Note that the scalar dissipation rate χ is different than the

turbulent dissipation rate ε that is a turbulent flow property and equal to the energy transfer

rate from the large eddies to the smaller eddies. χ is an important quantity in a turbulent

non-premixed combustion to describe a mixing rate. Chemical effects are incorporated

through the flame structure in mixture fraction space. The flamelet model (Peters, 2000)

expresses the mean reaction rate using a PDF formulation according to

50

( ) ( ) ( )1

0 0,l l st st stz f z f dzdω ω χ χ χ

∞= ∫ ∫ . (2.85)

( )f z presumes to be -functionsβ and stχ is often assumed to follow a log-normal

distribution. The laminar reaction rate ( ),l stzω χ needs to be calculated and stored in a

library before a combustion simulation is performed.

The eddy dissipation concept model (Magnussen and Hjertager, 1977; Magnussen, 1981) is

an extension of the eddy breakup model to non-premixed combustion. The mean fuel

burning rate is estimated from the following expression

( )

min , ,1

O PF m F

Y YC Yk s sεω ρ β

⎛ ⎞≈ ⎜ ⎟⎜ ⎟+⎝ ⎠

. (2.86)

mC and β are model constants. FY , OY , and PY are the mean mass fractions of fuel,

oxidizer, and products, respectively. s is the reaction coefficient for the fuel. The eddy

dissipation concept was further used to formulate a species conversion rate based on the

characteristic time in the reacting flow (Kong et al., 1995). The characteristic-time model

considers the influence of both laminar chemistry and turbulent mixing in determining the

overall reaction rate.

Closure models in LES approach

For premixed flames, the LES approach encounters a difficulty in resolving a very thin flame

front which is entirely on the subgrid scale (Pitsch, 2006). This was addressed by using an

artificially thickened flame (Colin et al., 2000), or a flame front tracking technique, i.e.,

level-set method ( -equationG ) (Pitsch, 2005), or filtering with a Gaussian filter larger than

51

the mesh size (Boger et al., 1998). A simple approach is to extend RANS models to LES.

For instance, the eddy breakup model can be rewritten as

( ),

1 1Let SGS

Cθω ρ θ θτ

= − (2.87)

where LeC is a constant, θ is a progress variable and θ is its resolved mean, and ,t SGSτ is a

subgrid turbulent time scale

, 1/ 2t SGSSGSk

τ Δ≈ (2.88)

where Δ is the filter size and SGSk is the subgrid turbulent kinetic energy.

For non-premixed flames, the PDF concept can be extended from RANS to LES for both

infinitely fast chemistry and finite rate chemistry. In the infinitely fast chemistry, the

reaction rates are governed by the mixing. The filtered reaction rates depend only on the

mixture fraction z

( ) ( ) ( )1

0, , ,l lt z f z t dzω ω= ∫x x (2.89)

where ( ), ,f z tx is the subgrid scale PDF that may be either presumed or obtained by solving

a transport equation. In the finite rate chemistry, the species mass fraction depends on both

the mixture fraction and its scalar dissipation rate (Cook and Riley, 1998; Pitsch, 2006).

2.5 Adaptive Grid Methods

Adaptive mesh refinement (AMR) was initially developed to improve solution accuracy

when solving partial differential equations (Berger, 1982; Berger and Oliger, 1984). AMR

52

has then been extended for use in solving various types of equations for a wide variety of

engineering applications (Bell et al., 1994b; Pember et al., 1995; Berger and LeVeque, 1998;

Jasak and Gosman, 2000b; Wang and Chen, 2002; Anderson et al., 2004). The purpose of

using AMR was to simulate complex processes more accurately while controlling

computational cost. In general, there are three major adaptive methods: h-refinement, p-

refinement, and r-refinement (Jasak and Gosman, 2000a). The h-refinement adds grid points

in regions of high spatial activity and is popular in finite volume solvers due to its simplicity

without the need for grid redistribution. The p-refinement adjusts the local order of

approximation in appropriate regions of the domain and is popular in finite element method.

The r-refinement method does not change total grid points but redistributes the grid to

minimize approximation error.

AMR was also applied to improve engine spray simulation. Lippert et al. (2005)

incorporated local refinement algorithms into an in-house solver using the isotropic cell-

splitting approach for hexahedral elements. The criterion for controlling refinement and

coarsening was the sum of fuel vapor mass and droplet mass in each cell, normalized by the

total injected mass. A global error control method was also formulated to determine whether

the solution was good enough. To further alleviate the grid dependence of spray modeling,

an improved coupling of both the gas-to-liquid and liquid-to-gas parameters was also

performed. Numerical tests showed the effectiveness of AMR coupled with the improved

phase coupling technique in removing grid artifacts associated with spray modeling.

53

Bauman (2001) implemented a spray model into a CFD code that employed AMR. The goal

was to address the problem of insufficient solution resolution when a mesh with fixed cell

size was used in order to achieve reasonable run times in high-pressure spray simulations.

Tonini (2008) employed an adaptive local grid refinement methodology combined with a

calculation procedure that distributed the mass, momentum and energy exchange between the

liquid and gas phases. The adaptation could be performed on various unstructured meshes

such as tetrahedron, hexahedron, pyramid, and prism. The grid refinement of up to three

levels was tested. The numerical results showed that the proposed methodology offered

significant improvements in dense spray simulation compared to the standard Lagrangian

method.

2.6 Parallel Computing

Multidimensional engine modeling can be computationally intensive, especially when

detailed combustion models or advanced turbulence models are used. Serial computation can

be impractical for engineering application. High performance parallel computing using

multiple processors can greatly reduce clock time and benefit product development for

industry. Rapid advances of computer technologies and development in parallel algorithms

have enabled researchers to use massive parallel computation for complex problems.

There are two major parallel programming algorithms: shared-memory method and message-

passing method. The shared-memory method treats the total memory of the computer as

equally accessible to each processor but the access may be coupled by different bandwidth

54

and latency mechanism. For optimal performance, parallel algorithms must consider non-

uniform memory access. Programming in shared memory can be done in a number of ways,

some based on threads, others on processes. The thread-based method has some advantages.

Synchronization and context switches among threads are faster than among processes.

Creating additional thread of execution is usually faster than creating another process. Many

thread-based libraries are available among which OpenMP is the most popular. In the

message-passing method, each processor has its own memory which is accessible only to that

processor. Processors can interact only by passing messages. The most common form of

this method is the Message Passing Interface (MPI). MPI provides a mature, capable, and

efficient programming method for parallel computation. It is highly portable and also the

most common method for parallel computing.

There are numerous applications of parallel computation for engine simulation. Yasar et al.

(1995) developed a parallel version of KIVA-3 coupled with the use of a block-wise

decomposition scheme to ensure an efficient load balancing and low

communication/computation ratio. Filippone et al. (2002) parallelized KIVA-3 using BLAS

library based on the MPI method. Aytekin (1999) implemented KIVA-3 on a distributed-

memory machine based on one-dimensional domain decomposition using the MPI library for

large eddy simulation. Zolver et al. (2003) developed an OpenMP-based parallel solver in

the code KIFP for diesel engine simulation using unstructured meshes and reached a speed-

up of three on four processors. The most recent advance in the KIVA code development is

the parallel version of KIVA-4 using the MPI library (Torres and Trujillo, 2006). In this

parallel version, hydrodynamic calculations, spray, combustion, and piston movement are all

55

parallelized. Domain decomposition of an arbitrary unstructured mesh can be performed

automatically based on a graphic method by using a package called METIS (Karypis and

Kumar, 1998a). Dynamic domain re-partitioning during the course of a computation was

also implemented to further enhance parallel performance by addressing the dynamic change

of active cells due to the port deactivation and piston movement.

2.7 KIVA Code

Among various engine simulation codes, the KIVA code is the most widely used code in the

research community. KIVA is used as a base CFD code based on which various physical and

chemistry models are developed. The first version, KIVA, was released in 1985 which was

capable of computing transient compressible flows with fuel sprays and combustion in

relatively simple geometries (Amsden et al., 1985a; Amsden et al., 1985b). A later

improvement is KIVA-II (Amsden et al., 1989) that included implicit temporal differencing,

more accurate advection with an improved upwinding scheme, and a k ε− turbulence model.

KIVA-3 (Amsden, 1993) added the capability of using a block-structured mesh in which

multiple blocks of cells could be patched together to construct a more complex engine mesh.

The code was enhanced by an improved snapping procedure to remove or add layers of cells

during piston movement. KIVA-3V (Amsden, 1997) added algorithms to simulate moving

valves and also included a liquid wall film model. Various advanced sub-models for sprays,

turbulence, and combustion were developed and added to this version for engine analysis and

design by a significant number of institutions.

56

KIVA-4 (Torres and Trujillo, 2006), the latest version of the KIVA code, represents a

fundamental change in the numerics in order to accommodate unstructured meshes. The

unstructured meshes can include various cell types such as hexahedra, tetrahedral, prisms,

and pyramids. Another important improvement is that KIVA-4 was parallelized using MPI.

Additionally, a collocated version of KIVA-4 was also available for further development of

advanced numerical schemes such as local mesh refinement.

This study made use of different versions of the KIVA code to develop advanced models and

numerical algorithms. Because KIVA-3V is well established and validated, this version was

used as the base code for the development of an LES turbulence model. On the other hand,

due to its flexibility in handling unstructured mesh, the collocated KIVA-4 was used to

develop the adaptive mesh refinement algorithms for more efficient and accurate spray

simulation.

57

3 DIESEL ENGINE COMBUSTION MODELING

3.1 Introduction

This chapter describes the major models that were implemented into KIVA-3V (Amsden,

1997) for diesel spray combustion simulation in the context of LES. The previously

developed combustion model based on detailed chemical kinetics (Kong et al., 2007b) will

also be described for the completeness of this thesis. This chapter will be arranged as

follows. First, subgrid models will be presented for the modeling of gas-liquid two-phase

flows in the context of LES, followed by the description of combustion and emissions

models. Then, model validations will be performed in two cylindrical chambers by

comparing simulation results against experimental data. Finally, the present model will be

used to simulate diesel combustion in a heavy-duty diesel engine using the updated KIVA-

3V that consisted of LES turbulence, KH-RT spray breakup, and detailed chemistry models.

3.2 LES Turbulence Models

LES has traditionally focused on the modeling of sub-grid scale stress tensor. Various types

of models have been developed and results were satisfactory compared against experimental

data. For turbulent reacting flows with sprays, however, the model development and

validation seem to be far less satisfactory due in part to the lack of experimental data. One of

the goals of this chapter is to use the LES models to simulate diesel spray combustion.

58

The compressible LES equations and common closure models are already presented in

Chapter 2. Specific models used for this study are further discussed. Note that the filter

width is the local grid size. A box filter is used since it is suitable for finite volume method

that is used in the present code. It is also assumed that the filtering operation commutes with

differentiation, i.e., t tφ φ∂ ∂ = ∂ ∂ , x xφ φ∂ ∂ = ∂ ∂ , although this may not hold in a non-

uniform mesh (Pomraning and Rutland, 2002).

The filtered terms that require special attention for closure models in reactive flows include

the sub-grid scale stress tensor ijτ , sub-grid scale scalar fluxes j lu Yτ and

ju Tτ , the subgrid

particle-gas interactions si iF uτ , the filtered rate of momentum gain per unit volume due to

spray siF , and the filtered chemical source term c

lω .

The subgrid stress tensor ijτ is directly estimated by using the one-equation non-viscosity

dynamic structure model (Pomraning and Rutland, 2002), which rescales the Leonard stress

tensor ijL with both its trace kkL and the sub-grid scale kinetic energy k

2ij ij

kk

k LL

τ = . (3.1)

The sub-grid scale kinetic energy k is solved from a transport equation that is derived from

the filtered momentum equation as

si i

jt ij ij F u

j j j

u kk k St x x x

ν τ ε τ⎛ ⎞∂∂ ∂ ∂

+ = − − +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ (3.2)

59

where tν and ε are modeled from Eq. (2.59) to Eq. (2.60) in Chapter 2. si iF uτ is a new

unclosed term due to sub-grid gas-droplet interactions and is defined by

si i

s sF u i i i iu F u Fτ = − . (3.3)

Adopting the similar modeling approach to the source term used in the k ε− equations in the

RANS approach, this term is modeled by the dot product of aerodynamic drag acting on a

particle and turbulent fluctuation as follows (Menon and Pannala, 1997; Banerjee et al., 2009)

"si i

sF u i iu Fτ ≈ − (3.4)

where "iu is the sub-grid i velocity component which cannot be directly resolved and must be

modeled. In this study, an approximate deconvolution (Shotorban and Mashayek, 2005) is

used to reconstruct the instantaneous velocity from the resolved velocity field by using the

truncated Van Cittert series expansion as follows

( ) ( ) ( )*

01 * 2 ...

Nn

i i i i i i i in

u G u u u u u u u=

= − = + − + − + +∑ (3.5)

where *iu is the modeled instantaneous velocity component, iu , iu , iu are the filtered values

corresponding to the grid level, the second filter level, and the third filter level, respectively.

G is the filter kernel. Taking a 2nd order approximation from the above series expansion, the

modeled fluctuating component "iu can then be obtained from Eq. (3.5) as

( ) ( )" * 2i i i i i i i iu u u u u u u u= − = − + − + (3.6)

The filtered rate of momentum gain per unit volume due to spray siF is approximated by

using the resolved quantities as in RANS approach (Amsden et al., 1989).

60

For chemical species, the sub-grid scalar flux j lu Yτ is approximated by the gradient model as

j l

t lu Y

j

YSc xντ ∂

= −∂

(3.7)

where lY is the mass fraction of species l , Sc is the turbulent Schmidt number with a value

of 0.68 (Pomraning and Rutland, 2002). However, to be consistent with the non-viscosity

model concept used in closing the sub-grid stress tensor, a model that scales with the

Leonard-like term should usually be used to close the above sub-grid scalar flux.

The chemical source term clω and the spray source term s

lω are assumed to be equal to the

resolved scales. The filtered energy equation is expressed in terms of the resolved

temperature as

ju Tv j j j c sv i lij l

lj j j j j j j

c u T u qc T u Yp D h Q Qt x x x x x x x

ρτρρ σ ρ∂ ⎛ ⎞∂ ∂ ∂∂ ∂ ∂∂

+ = − − − + + + +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠∑

(3.8)

where cQ and sQ are source terms due to chemical reaction and spray, respectively, and are

modeled by the resolved scales. jq is the molecular heat flux and is approximated by

neglecting the sub-grid scale of thermal conductivity. vc is the total specific heat of the

mixture at constant volume. For low to moderate Mach number flow typical of engine in-

cylinder flows, the first and fourth terms are approximated as follows (Moin et al., 1991;

Vreman et al., 1994)

j j

j j

u up p

x x∂ ∂

≅∂ ∂

(3.9)

61

i iij ij

j j

u ux x

σ σ∂ ∂≅

∂ ∂. (3.10)

Similarly, the fifth term is approximated by neglecting the sub-grid scale effect as

l ll l

l lj j j j

Y YD h D hx x x x

ρ ρ⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂

=⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠∑ ∑ . (3.11)

The sub-grid heat flux term ju Tτ is defined as

( )ju T v j jc u T u Tτ = − (3.12)

and is modeled by the gradient method analogous to the sub-grid scalar fluxj mu Yτ as

Prj

p tu T

t j

c Tx

ντ ∂

= −∂

(3.13)

where Prt is the turbulent Prandtl number with the value of 0.9 (Amsden et al., 1989). pc is

the total specific heat of the mixture at constant pressure. The calculations of pc and vc are

taken as the summations of mass-weighted value of each individual species (Amsden et al.,

1989).

As discussed in Chapter 2, spray modeling has been shown to be dependent on the grid

resolution due to a number of reasons. One of the reasons is that the representative gas phase

flow quantities at the particle location cannot be directly solved due to the use of finite grid

points. To improve the gas-to-liquid momentum exchange, the gas phase velocity at the

particle location is taken as a weighted value of the velocity of each node of the cell

containing the particle by using an interpolation scheme (Nordin, 2001) as

62

8

, ,1

g p n g nn

w=

=∑u u (3.14)

where ,g pu is the gas phase velocity at the particle location, ,g nu is the nodal gas phase

velocity, and nw is the weight associated with the distance between each node and the

particle by

2,

82

,1

n pn

i pi

lw

l

−

−

=

=

∑ (3.15)

where ,i pl indicates the distance between the node i and the particle p .

3.3 Combustion and Emissions Models

Engine combustion can involve hundreds of species and thousands of reactions with a wide

spectrum of chemical time scales. Different combustion modes can also have different

critical events. For instance, ignition and mixing are significant processes for diesel

combustion whereas flame propagation is important for gasoline combustion. This section

will describe the chemical kinetics used to simulate diesel spray ignition, combustion, and

emissions formation.

In the current combustion model, the sub-grid scale turbulence-chemistry interactions are not

considered and the mean reaction rate is assumed to be controlled mainly by the kinetics.

Nonetheless, turbulence influences chemical reactions through species transport and mixing

at the grid level. The chemical reactions are directly solved at each computational time step.

63

The rate of change of mass fraction for a species l is given by treating each computational

cell as a chemical reactor (Kong et al., 2007b)

l l ldY Wdt

ωρ

= (3.16)

where lY is the mass fraction of species l , lω is the concentration production rate, lW is the

molecular weight of species l , and ρ is the total density. Assuming a constant cell volume,

the energy equation can be written as

1

0L

v l l ll

dTc e Wdt

ρ ω=

+ =∑ (3.17)

where vc is the specific heat of the mixture at constant volume, T is the cell temperature,

and le is the specific internal energy of species l . In the context of LES, ρ , T , and lY are

the resolved quantities.

Diesel fuel is a mixture composed of approximately 200 to 300 hydrocarbon species. A

comprehensive reaction mechanism for practical diesel fuel is not available. In this study, a

skeletal reaction mechanism for n-heptane oxidation was used to simulate diesel fuel

chemistry due to their similar ignition characteristics and cetane numbers (Patel et al., 2004).

The resulting mechanism retained the main features of the detailed mechanism and included

reactions of polycyclic aromatic hydrocarbons. Additionally, a reduced NO mechanism was

obtained by using the same reduction methodology based on the Gas Research Institute (GRI)

NO mechanism (Smith et al., 2000). The resulting NO mechanism contained only four

additional species (N, NO, NO2, N2O) and nine reactions that describe the formation of nitric

oxides.

64

The NO mechanism was incorporated into the mechanism for n-heptane oxidation to form a

skeletal reaction mechanism for the diesel fuel chemistry used in this study. As a result, the

reaction mechanism for fuel oxidation and NOx emissions consisted of 34 species and 74

reactions (Kong et al., 2007b). The CHEMKIN chemistry solver was implemented into

KIVA in order to use the above reaction mechanism for diesel combustion simulation.

The capability to predict the engine-out soot emissions is important for a CFD model, and the

predicted engine-out soot can also provide the boundary condition for after-treatment

modeling. A phenomenological soot model (Han et al., 1996) was used to predict soot

emissions in this study. Two competing processes, soot formation and soot oxidation, were

combined to determine the rate of change of soot mass as

sfs sodMdM dMdt dt dt

= − (3.18)

where sM , sfM , soM are the masses of the soot, soot formation, and soot oxidation,

respectively. The formation rate uses an Arrhenius expression and the oxidation rate is based

on a carbon oxidation model, described respectively as

2 2C H expsf sfn

sf

dM EA M P

dt RT⎛ ⎞

= −⎜ ⎟⎝ ⎠

(3.19)

6so cs Total

s s

dM Mw M Rdt Dρ

= (3.20)

where 2 2C HM , cMw , P , T , TotalR , R are acetylene mass, soot molecular weight, pressure,

temperature, surface mass oxidation rate, and the universal gas constant, respectively. sρ

65

and sD are soot density and average particle diameter. The soot formation rate uses

acetylene (C2H2) as the inception species in Eq. (3.19) since acetylene is the most relevant

species pertaining to soot formation in the present reaction mechanism (Kong et al., 2007a).

The soot oxidation rate is determined by the Nagle-Strickland-Constable model that

considers carbon oxidation by two reaction pathways whose rates depend on the surface

chemistry of two different reactive sites (Han et al., 1996). Model constants are 150,sfA =

52,335 Joule ,-1sfE mol= ⋅ 2 ,-3

s g cmρ = ⋅ and 62.5 10sD cm.−= × In the present calculation,

acetylene was assumed to form soot particles which, in turn, were converted to CO, CO2 and

H2 as a result of oxidation. Again, in the context of LES, the above equations used the

resolved quantities.

3.4 Spray Combustion Modeling

Diesel spray simulations were performed in two constant-volume chambers to validate the

LES models. First, a non-evaporating diesel spray was simulated using both the RANS and

LES approaches. Then, diesel spray combustion in a combustion chamber was simulated

using both LES and RANS approaches to demonstrate the capability of LES. Note that spray

sub-models developed in the context of the RANS approach were retained in LES

simulations except that all the RANS-based scales were replaced by the LES-based scales.

For instance, the turbulent kinetic energy was replaced by the subgrid scale kinetic energy,

and the turbulent dissipation rate was replaced by the subgrid kinetic energy dissipation rate.

66

Among the sub-models, the fuel spray atomization was simulated by modelling the growth of

unstable surface waves on the liquid surface that will result in drop breakup (Patterson and

Reitz, 1998; Kong et al., 1999). In the combustion case, combustion chemistry was

simulated by using a skeletal n-heptane mechanism for fuel oxidation and NOx emissions that

consisted of 34 species and 74 reactions. The phenomenological soot model discussed in the

previous section was used to predict soot emissions. Other sub-models included those for

spray/wall interaction, wall heat transfer, and piston-ring crevice flow. Before the

development of a good wall model for LES, the present approach used the RANS-based wall

function, i.e., the turbulent law-of-the-wall conditions and fixed temperature walls. This was

also to prevent the use of very fine mesh in the near-wall region. Thus, the present practice

belongs to a hybrid RANS/LES approach, as often referred to Very Large Eddy Simulation

(VLES) (Pope, 2000).

Non-evaporative spray in a constant chamber

A non-evaporating diesel spray in a constant-volume chamber was simulated using the LES

approach to validate the LES models. Predicted spray structure and liquid penetration will be

compared with experimental data (Dan et al., 1997; Hori et al., 2006). The LES results will

also be compared with RANS results in terms of liquid penetration, spray structure, velocity

vector, and vortex structure. It should be noted that instantaneous LES results should be

ensemble- or time-averaged to compare to ensemble-averaged experimental results. However,

due to the significant time required for the averaging, the instantaneous LES results were

used for comparison. In the RANS approach, RNG k ε− model was used whereas in the LES

approach, the subgrid kinetic energy was solved from the transport equation. Spray was

67

injected at the center of the top surface along the chamber axis. All simulations had the same

computational duration, i.e., started from zero and ended at 4 ms. Table 3.1 lists the

simulation conditions.

Table 3.1. Simulation conditions for a non-evaporative spray.

Fuel n-C13H28 Injection duration 1.8 ms Injection profile Top-hat Injection orifice diameter 200 µm Orifice pressure drop 77 MPa Fuel mass 0.012 g Fuel temperature 300 K Ambient temperature 300 K Ambient pressure 1.5 MPa Ambient density 17.3 kg/m3

Ambient gas N2 Number of computational parcels 2000 Bore (cm) × stroke (cm) 3.0 × 10.0 Cell division number (radial × azimuthal × axial)

8×60×50, 15×60×100, 30×60×200

Spray models All improved sub-models excluding evaporation model

Turbulence model LES subgrid models and RNG - k ε−

To test the effect of grid resolution on simulation results, three different mesh sizes were

used, as shown in Figure 3.1, which corresponded to the division numbers of 8×60×50,

15×60×100, and 30×60×200 in radial, azimuthal and axial directions, or approximately an

averaged radial cell size of 2 mm, 1 mm, and 0.5 mm, respectively. Meanwhile, all meshes

were created as cylindrical meshes since this type of mesh has been shown to be able to

reduce the grid dependence of spray simulation due to a better resolution in the azimuthal

direction (Hieber, 2001; Baumgarten, 2006).

68

Figure 3.1. Three mesh sizes used in non-evaporative spray simulations in a constant volume

chamber (top view, injector is located at the center).

No-slip boundary was enforced for velocity, and fixed temperature was enforced for wall

temperature. For other scalars, zero fluxes were enforced at the wall boundaries. Initial gas

velocity was set to zero, and initial temperature and pressure were set to the values as in the

experiments. For the RANS approach, initial k was set to the given input value and initial ε

was calculated from k and the given integral length scale (Amsden et al., 1989). For the

subgrid kinetic energy in LES, its initial value was given a negligibly small value

(Chumakov, 2005).

Figure 3.2 shows the history of the liquid penetration using different mesh sizes in the LES

simulations. Note that the liquid penetration is defined as the axial distance from the nozzle

orifice to the location that corresponds to 95% of integrated fuel mass of total injected fuel

from the orifice. It is seen that the predicted liquid penetrations agree better with

experimental data as the mesh size is reduced due mainly to the fact that LES can resolve

69

more flow structures for the finer mesh. The liquid penetrations using the finest mesh (0.5

mm) match very well with experimental data (Dan et al., 1997; Hori et al., 2006). It is also

seen that spray penetration is still dependent on the grid resolution although an interpolation

method was used to obtain a weighted gas velocity at the particle location in order to reduce

this dependence. Further improvements to the liquid-to-gas coupling terms and the drop

collision model may be needed in order to eliminate this grid dependence. Figure 3.3 shows

the liquid penetrations using different mesh sizes in the RANS simulations. The simulated

penetrations also match quite well with experimental data for the finest grid size (0.5 mm).

However, LES predicted overall better penetrations than RANS in the entire history due to

the fact that LES can resolve large turbulent eddies and only need to model subgrid eddies,

whereas RANS needs to model all turbulent eddies.

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.0 1.0 2.0 3.0 4.0Time (ms)

Liq

uid

Pene

trat

ion

(cm

)

Non-evap-LES-2 mmNon-evap-LES-1 mmNon-evap-LES-0.5 mmMeasurement (Dan et al., 1997)

Figure 3.2. Spray liquid penetrations using different grid sizes in LES simulations.

70

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.0 1.0 2.0 3.0 4.0Time (ms)

Liq

uid

Pene

trat

ion

(cm

)

Non-evap-RANS-2 mmNon-evap-RANS-1 mmNon-evap-RANS-0.5 mmMeasurement (Dan et al., 1997)

Figure 3.3. Spray liquid penetrations using different grid sizes in RANS simulations.

Measurement at t=1.8 ms

0.5 mm 1.0 mm 2.0 mm

Figure 3.4. Spray structure using different grid sizes in LES simulations at 1.8 ms

(measurement by Dan et al., 1997).

71

Measurement at t=1.8 ms

0.5 mm 1.0 mm 2.0 mm

Figure 3.5. Spray structure using different grid sizes in RANS simulations at 1.8 ms

(measurement by Dan et al., 1997).

Figure 3.4 and Figure 3.5 show spray structures using different mesh sizes in the LES and

RANS simulations, respectively. As the grid size was reduced, LES predicted a more

realistic spray structure than RANS by comparing the predicted drop distributions with the

experimental image. In particular, the LES approach using the finest mesh could predict the

dynamic structure that was not observed in the RANS results. For the finest mesh, both LES

and RANS predicted some overly dispersed particles at the leading edge of the spray. The

differences of the spray structure between LES and RANS can also be seen from the gas

velocity vector on a central cutplane (Figure 3.6) and the 3-D vortex structure (Figure 3.7).

The vortex structure was visualized by using a positive value of the second invariant

12

ji

j i

uuQ x x∂⎛ ⎞∂

= −⎜ ⎟∂ ∂⎝ ⎠ of the velocity gradient tensor, the so-called criterion,Q − which can be

used to indicate resolved coherent structures (Hunt et al., 1988; Dubief and Delcayre, 2000;

72

Fujimoto et al., 2009). Alternatively, Q indicates a relative movement between rotation and

deformation (Dubief and Delcayre, 2000) and can be defined as

( )12 ij ij ij ijQ S S= Ω Ω − (3.21)

where ijΩ and ijS are the vorticity tensor and the rate of strain tensor, respectively, and

defined as follows

1 1,2 2

j ji iij ij

j i j i

u uu uSx x x x

⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂Ω = − = +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

. (3.22)

Thus, a positive value of Q represents vortex. The critical value of Q , which gives the

highest level of vorticity fluctuations, is determined based on the characteristic parameters as

(Domingo et al., 2006)

2

injc

nozzle

VQd

= (3.23)

where injV is the average injection velocity, and nozzled is the nozzle diameter. Figure 3.7

shows an iso-surface of Q corresponding to its critical value of 610cQ = for both RANS and

LES simulations. The LES result shows a clear 3-D coherent vortex tubes which are not

equally observed in the RANS result due to the nature of its statistical average and entirely

modeled turbulence. Although the coherent vortex structure can be predicted by the current

LES approach, the evolution of this structure may have been suppressed because the current

convective calculation using the quasi-second order upwind scheme may result in numerical

diffusion (Hori et al., 2008). Low numerical diffusion scheme may be needed for the

convection calculation in LES simulation (Fujimoto et al., 2009).

73

Figure 3.6. Velocity vector on a central cutplane (y=0) in both RANS and LES simulations at

1.8 ms.

Figure 3.7. Vortex structure plotted as an iso-surface of the critical value cQ of 1.0e+6 in both

RANS and LES simulations at 1.8 ms.

74

Diesel spray combustion in a constant-volume chamber

The purpose of the present modeling was to demonstrate the capability of LES to predict the

dynamic flame structure and soot distribution in a diesel combustion flame. The flame

structure of diesel spray in a constant-volume combustion chamber was studied in Sandia

National Laboratory using laser diagnostics (Pickett and Siebers, 2004; Siebers and Pickett,

2004). The details of the flame structure have provided new insights into the combustion and

emission formation processes in diesel sprays and provided data for model validation (Kong

et al., 2003; Tao et al., 2004). For instance, a diesel flame consists of a diffusion zone at the

fuel jet periphery, where NOx is formed, and a fuel rich reaction zone in the center of fuel

spray, where the majority of soot is formed (Kong et al., 2007a). Additionally, the flame lift-

off length will determine the time available for fuel-air mixing prior to ignition and entering

the reacting zone and thus will affect the sooting tendency of the diesel fuel jet. The

numerical simulation of the diesel flame using the RANS approach coupled with detailed

chemistry gave satisfactory results in global quantities such as the lift-off length, mean

cylinder pressure, heat release rate, and soot and NOx emissions (Kong et al., 2007a).

However, due to its statistical nature, the RANS approach was not able to predict the

dynamic flame structure which can be important to the prediction of the soot and NOx

distributions. This study was to test the capability of the current LES approach to simulate a

diesel flame when coupled with detailed chemistry and a phenomenological soot model.

Experimental data of the Sandia diesel combustion chamber were used to validate the current

LES models. The baseline experimental conditions were list in Table 3.2. The

computational domain was 6.3 cm in diameter and 10.0 cm in height. The computations used

75

a cylindrical mesh with an average grid size of 2 mm in both radial and axial directions. The

spray was injected from the top surface and directed downward along the cylinder axis.

Spray dynamics were modeled by the improved sub-models as discussed previously.

Combustion chemistry was simulated by using a skeletal n-heptane mechanism for fuel

oxidation and NOx emissions that consisted of 34 species and 74 reactions, while soot

emissions was modeled using a phenomenological model. Similar boundary and initial

conditions to the non-evaporative spray case were applied. RANS simulations using the

RNG k ε− turbulence model were also performed for comparison.

Table 3.2. Experimental conditions for model validation.

Fuel #2 diesel Injection system Common-rail Injection profile Top-hat Injection orifice diameter 100 µm Orifice pressure drop 138 MPa Discharge coefficient 0.8 Fuel temperature 436 K Ambient temperature 1000 K Ambient pressure 4.18 MPa Ambient density 14.8 kg/m3

O2 concentration 21%

Figure 3.8 shows the comparison of soot distributions on a central cutplane among RANS,

experimental, and LES results. Figure 3.9 shows the comparison of soot iso-surfaces at

different times. The experimental images were obtained from planar laser induced

incandescence (PLII) that indicated the soot mass along the laser path (Pickett and Siebers,

2004). It can be seen that LES predicted both the dynamic structure and lift-off length closer

to the experiment than those by RANS which mostly predicted an axisymmetric soot

76

distribution. The different soot distributions predicted by LES and RANS indicated the

differences in local flow parameters predicted by the two approaches. These local flow

parameters were directly linked to the local equivalence ratio which was largely determined

by the turbulent mixing. The LES can resolve more flow structures and thus more dynamic

soot structures can be seen in the LES results. However, LES seems to over-predict the fuel

vapor penetration resulting in an over-extended reaction zone. Further improvements in LES

for predicting evaporating sprays are required.

0 20 40 60 80 100

RANS

0 20 40 60 80 100

LESMeasurement

Dashed lines indicated the lift-off length Figure 3.8. Comparison of soot distributions at 1.3 and 1.7 ms after start of injection for

dnozzle=100 µm, Tamb=1000 K, ΔPamb=138 MPa, ρamb=14.8 kg/m3 (measurement by Pickett and Siebers, 2004).

RANS

0 20 40 60 80 100

LESMeasurement

0 20 40 60 80 100 Figure 3.9. Comparison of soot iso-surface predicted by RANS and LES and the

experimental PLII images (Pickett and Siebers, 2004) for dnozzle=100 µm, Tamb=1000 K, ΔPamb=138 MPa, ρamb=14.8 kg/m3.

77

To further demonstrate the distinct flame structures predicted by the LES and RANS

approach, Figure 3.10 shows the comparison of temperature at different times. Although no

experimental data were available for comparison, a similar conclusion as in the soot analysis

can be drawn that LES predicted a more dynamic, and likely more realistic, reaction zone

structure. Additionally, LES predicted thinner diffusion flame which is consistent with

experimental observation of NOx distribution (Dec, 1997) since thermal NO is very sensitive

to temperature above 1900 K (Law, 2006).

0 20 40 60 80 100

1.3

1.7

0 20 40 60 80 100

LESRANS

Figure 3.10. Comparison of predicted temperature distributions using RANS and LES for

dnozzle=100 µm, Tamb=1000 K, ΔPamb=138 MPa, ρamb=14.8 kg/m3.

3.5 Diesel Engine Simulation

The present model was also applied to simulate the in-cylinder spray combustion process in a

diesel engine. As in the spray simulations in the previous section, the fuel spray atomization

was simulated by modelling the growth of unstable surface waves on the liquid surface that

will result in drop breakup (Patterson and Reitz, 1998; Kong et al., 1999). Combustion

78

chemistry was simulated by using a skeletal n-heptane mechanism for fuel oxidation and

NOx emissions that consisted of 34 species and 74 reactions. The phenomenological soot

model (Han et al., 1996) was used to predict soot emissions. Other sub-models included

spray/wall interactions, wall heat transfer, and crevice flows. The turbulent law-of-the-wall

conditions and fixed temperature walls were enforced near solid walls.

Table 3.3. Engine specifications and operating conditions.

Engine model Caterpillar 3401 SCOTE Bore × stroke 137.2 mm × 165.1 mm Compression ratio 16.1:1 Displacement 2.44 Liters Connecting rod length 261.6 mm Squish height 1.57 mm Combustion chamber geometry In-piston Mexican hat with sharp edged crater Piston Articulated Charge mixture motion Quiescent Injector HEUI Maximum injection pressure 190 MPa Number of nozzle holes 6 Nozzle hole diameter 0.214 mm Included spray angle 145˚ Injection rate shape Rising Experimental conditions for model validations (Klingbeil et al., 2003) Case group SOI (ATDC) A (8% EGR) -20, -15, -10, -5, 0, +5 B (27% EGR) -20, -15, -10, -5, 0, +5 C (40% EGR) -20, -15, -10, -5, 0, +5

The specifications of the engine are listed in Table 3.3 (Li and Kong, 2008). The engine

operating conditions were optimized to achieve low NOx and particulate emissions by

varying the start-of-injection (SOI) timing and exhaust gas recirculation (EGR). The

experimental results indicated that low emissions could be achieved by optimizing the

79

operating conditions to allow an optimal time interval between the end of fuel injection and

the start of combustion. This is to allow a longer mixing time for a more homogeneous

mixture. In this study, the cylinder pressure history and exhaust soot and NOx emissions were

used for model validations.

Injector

Piston movingdirection

Figure 3.11. Computational mesh (a 60° sector mesh).

The computation used a 60-degree sector mesh that included a full spray plume since the

injector had six nozzle holes that were uniformly oriented in the circumferential direction.

The injector was located at the top of the domain as shown in Figure 3.11. The average grid

size was approximately 2 mm and the number of computational cells was approximately

20,000 at bottom-dead-center. The cylindrical mesh was finer near the injector for fuel spray

simulations. The present mesh resolution was considered to be adequate for engine modeling

and has been used in RANS simulations in previous studies (Kong et al., 2007b; Kong et al.,

80

2007a). A mesh sensitivity study of using the present LES model also showed that the

predicted global properties, i.e. cylinder pressures, match well between a standard mesh and

a fine mesh with four times of the standard mesh number (Jhavar and Rutland, 2006). The

current mesh was adequate for the present engineering application since the model will be

validated by comparing the global parameters such as cylinder pressures and soot and NOx

emissions.

Simulations started from intake valve closure (IVC) with a swirl ratio of 1.0 and a uniform

mixture of air and EGR was specified. Wall temperature boundary conditions used 433 K for

the cylinder wall, 523 K for the cylinder head and 553 K for the piston surface. Computations

ended at exhaust valve open (EVO) when predicted soot and NOx emissions were compared

with engine exhaust measurements.

Operating conditions listed in Table 3.3 were simulated with the same set of model constants

and kinetics parameters. In general, the model performed well in predicting combustion for

the range of conditions studied. Figure 3.12 shows the comparisons in cylinder pressure

histories and heat release rate data for 8% EGR cases. The present model predicted correct

ignition timing and combustion phasing. Note that similar levels of agreement between

measurements and predictions were also obtained for the other cases (not shown here for

brevity). A large portion of premixed burn was observed under the present engine conditions,

as also predicted by the model. Although the present mesh is not as fine as those usually used

in LES modeling, the possible non-equilibrium effects (i.e. the filter cutoff is in the inertial

81

range) are reasonably accounted for by the use of the transport equation for the sub-grid

kinetic energy.

0

1

2

3

4

5

6

7

8

9

-60 -40 -20 0 20 40 60

Crank Angle (ATDC)

Pres

sure

(MPa

)

0

600

1200

1800

2400

3000

3600

4200

4800

5400

Hea

t Rel

ease

Rat

e (J

/deg

)-20-10

+5

0

1

2

3

4

5

6

7

8

9

-60 -40 -20 0 20 40 60

Crank Angle (ATDC)

Pres

sure

(MPa

)

0

600

1200

1800

2400

3000

3600

4200

4800

5400

Hea

t Rel

ease

Rat

e (J

/deg

)-20-10

+5

Figure 3.12. Comparisons of measured (solid lines) (Klingbeil et al., 2003) and predicted

(dashed) cylinder pressure and heat release rate data for SOI = –20, –10 and +5, both with 8% EGR.

Soot and NOx are two major pollutants of diesel engines and are difficult to be predicted

accurately over a wide range of operating conditions. Predictions of soot and NOx emissions

strongly depend upon the model accuracy in predicting both the overall combustion and the

local mixture conditions. Figure 3.13 shows the history of the in-cylinder soot and NOx

mass compared with the engine-out exhaust measurements. It is seen that NOx chemistry

freezes during expansion due to decreasing gas temperature. The history of soot mass

indicates that soot is formed continuously at the beginning and a large amount of soot is

oxidized later in the engine cycle.

82

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

-60 -10 40 90 140

Crank Angle (ATDC)

NO

x(g/

kgf)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Soot

(g/k

gf)

NOx

Soot

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

-60 -10 40 90 140

Crank Angle (ATDC)

NO

x(g/

kgf)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Soot

(g/k

gf)

NOx

Soot

Figure 3.13. Evolutions of predicted in-cylinder soot and NOx emissions compared with

engine exhaust measurements (solid symbols) (Klingbeil et al., 2003) for SOI = –5 ATDC, 40% EGR.

Figure 3.14 to Figure 3.16 show comparisons of predicted and measured soot and NOx

emissions. The trend and magnitude of NOx emissions are well predicted, indicating that the

chemical kinetics used in this study is adequate for the present conditions including the low

temperature combustion regimes at high EGR levels. It is known that NOx formation is

characterized by slow chemistry and sensitive to local temperatures. Model results indicate

that the instantaneous local species concentrations are reasonably modeled by the present

LES approach without considering the sub-grid scale effect of chemical reactions.

83

0

20

40

60

80

100

120

140

160

-25 -20 -15 -10 -5 0 5 10

SOI (ATDC)

NO

x (g

/kgf

)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Soot

(g/k

gf)

NOx (LES)

NOx (Exp)

Soot (LES)

Soot (Exp)

Figure 3.14. Comparisons of measured (Klingbeil et al., 2003) and predicted soot and NOx

emissions for 8% EGR cases.

0

10

20

30

40

50

60

70

80

90

100

-25 -20 -15 -10 -5 0 5 10

SOI (ATDC)

NO

x (g

/kgf

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Soot

(g/k

gf)

NOx (LES)NOx (Exp)Soot (LES)Soot (Exp)

.

Figure 3.15. Comparisons of measured (Klingbeil et al., 2003) and predicted soot and NOx

emissions for 27% EGR cases.

84

0

5

10

15

20

25

30

35

40

45

50

-25 -20 -15 -10 -5 0 5 10SOI (ATDC)

NO

x (g

/kgf

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Soot

(g/k

gf)

NOx (LES)NOx (Exp)

Soot (LES)Soot (Exp)

Figure 3.16. Comparisons of measured (Klingbeil et al., 2003) and predicted soot and NOx

emissions for 40% EGR cases.

The trends of soot emissions with respect to SOI and EGR are predicted reasonably well

except for the early injection cases for 27% EGR. In general, early fuel injection (e.g. –20

ATDC) will result in high combustion temperatures that enhance soot oxidation and lower

exhaust soot emissions. As the injection timing is retarded toward top-dead-centre (TDC),

soot emissions increases due to poor oxidation. As the injection timing is further retarded

passing TDC, in-cylinder gas temperature decreases noticeably, resulting in significantly

lower soot formation rates and final soot emissions. The model is able to predict the peak of

soot emissions occurring at SOI near TDC for various EGR levels.

It is of interest to demonstrate the unsteadiness of in-cylinder flows predicted by the present

LES approach and compare the flow structure with that predicted by the RANS models.

85

Figure 3.17 shows the fuel spray and temperature distributions for SOI = –20 ATDC with

8% EGR. It is seen that the LES model is able to predict the unsteady flow structure during

the engine process. Differences in CO mass fraction distributions predicted by both LES and

RANS are also observed, as shown in Figure 3.18. It can be seen that LES models can

indeed capture more detailed flow structures and can be further developed into a tool to

address variations due to subtle changes in engine operating conditions as well as cycle-to-

cycle variations.

Figure 3.17. Spray and temperature distributions on cross sections through spray at –14

ATDC for SOI = –20 ATDC with 8% EGR.

86

Figure 3.18. Distributions of CO mass fraction on a cross section through spray at –14 ATDC

for SOI = –20 ATDC with 8% EGR.

The present models performed well in predicting the overall performance of the engine

including the cylinder pressure history, heat release rate data, and soot and NOx emissions

trends with respect to injection timing and EGR levels. The present LES approach could also

predict the unsteadiness and more detailed flow structures as compared to RANS models.

Thus, the LES models can be further developed into an advanced engine simulation tool to

address issues such as cycle-to-cycle variations and to capture performance variations due to

the subtle change in engine operating conditions or geometrical designs.

87

4 GASOLINE SPRAY MODELING USING ADAPTIVE MESH

REFINEMENT

4.1 Introduction

This chapter describes the development and implementation of parallel adaptive mesh

refinement into KIVA-4 for gasoline spray modeling due to the capability of KIVA-4 to

handle unstructured meshes. Adaptive mesh refinement (AMR) can be used to increase the

grid resolution in the spray region to improve the spray simulation. This chapter will be

arranged as follows. First, a procedure of performing local grid refinement and coarsening is

presented, followed by the discussion of the modifications of hydrodynamic calculations to

accommodate AMR in KIVA-4. Second, the issues of implementing parallel AMR into

KIVA-4 and our strategies for addressing these issues are described. Third, both serial and

parallel AMR will be validated by performing gasoline simulations on three different engine

geometries. Finally, the AMR calculations in real engine geometries will be demonstrated.

4.2 Adaptive Mesh Refinement

It is known that the discrete Lagrangian particle method used in KIVA is grid dependent.

Adaptive mesh refinement (AMR) was implemented into KIVA-4 to increase spatial

resolution in the spray region to improve spray simulation. The standard KIVA-4 adopts a

“staggered” approach for solving momentum equations. To simplify numerical schemes and

the implementation procedure for AMR, this study adopted a “collocated” approach in which

velocity was solved at the cell center for the momentum equations and used for the gas-liquid

88

coupling terms. A pressure correction method proposed by Rhie and Chow (Tsui and Pan,

2006) was used to address unphysical pressure oscillations due to the collocation of pressure

and velocity (Ferziger and Peric, 2002). The current implementation determines data

structure and numerical methods for AMR based on the features of the KIVA-4 solver. For

instance, new child cells are attached to the existing cells. Cell edges from the higher-level

cells are attached to the lower-level cells at the coarse-fine interface to calculate edge-

centered values since the edge-looping algorithm is used to calculate diffusive fluxes, as will

be described in the calculation of diffusive fluxes.

The current adaptation increases the grid density by splitting a cell into eight child cells.

This method provides flexibilities in the mesh construction and is consistent with the

characteristics of the finite volume solver with the arbitrarily unstructured mesh (Jasak and

Gosman, 2000a). The adaptation starts with an initial coarse mesh (level 0) and creates new

grids with higher levels (level l ) continuously as the computation progresses. Meanwhile,

the refined grids will be coarsened to the lower level grids if the fine grid is not needed in

order to reduce the computational cost.

In this study, the adaptation is performed on a hexahedral mesh that is commonly used in

engine simulations. The adaptation criteria are based on normalized fuel mass and fuel vapor

gradients as will be described shortly. If a cell meets the criteria, it is tagged and will be

divided into eight child cells as shown in Figure 4.1 (a). If all the child cells of a parent cell

meet the coarsening criteria, all the child cells of the parent cell are de-activated and the

parent cell is restored. The detailed procedures are described as follows.

89

2

Level 1

Level 0131

2

(a) (b) Figure 4.1. Schematic of local mesh refinement: (a) a parent cell is divided into eight child

cells; (b) conventional face interface (1-3) and coarse-fine interface (1-2).

The refinement involves grid creation, connectivity setup, and property update. The

procedure consists of the following steps.

(1) Insert new vertices on each edge, face, and cell by interpolation to create new sub-edges,

sub-faces, and sub-cells;

(2) Establish relationship between the parent cell (level- ( )1l − ) and its direct child cells

(level- l );

(3) Deactivate the parent cell, and update the connectivity;

(4) Re-associate spray particles from the parent cell to the corresponding child cells based

on the shortest distance between a particle and the center locations of the child cells;

(5) Determine the cell-centered properties of the child cells using the linear variation with

local conservation.

For instance, a generic variable cQ defined on the cell center of a child cell can be obtained

by the linear variation of the function within the parent cell as

( ) ( )c p pQ Q Q= + − ∇ic px x (4.1)

90

where pQ is the value at the parent cell, ,c px x are the cell-center locations of the child cell

and parent cell, respectively. ( ) pQ∇ is the gradient evaluated on the parent cell using the

least-squares method.

If all the child cells of a parent cell meet the coarsening criteria, all these child cells are de-

activated and the parent cell is restored. The coarsening procedure consists of the following

steps.

(1) Update the connectivity data;

(2) Re-associate spray particles in the child cells to their parent cell;

(3) Determine the properties of the parent cell from the child cells by volume average or

mass average satisfying conservation laws.

The adaptation criterion uses a combination of “normalized fuel mass” and “vapor mass

fraction gradients.” The normalized fuel mass is the ratio of total mass ( l vm + ) of liquid and

vapor in a cell to total injected fuel mass ( injm ) as

l vm

inj

mr m+= . (4.2)

Because a majority of the injected fuel is still in liquid form, this criterion can ensure the

proper adaptation in the region near the injector nozzle. The vapor gradients are also chosen

in order to provide adequate grid resolution outside the spray periphery where the vapor

gradients are high. First derivatives (Wang and Chen, 2002) are used to calculate the

indicator from the vapor gradients as,

91

( ), ,c j jc jQ lτ = ∇ Δ (4.3)

where ( ) ,c jQ∇ is the cell-centered gradient component of the variable Q with respect to j

coordinate and jlΔ is the cell size in j direction. Note that the repeated index does not

imply Eisenstein summation. The following conditions are used for the grid adaptation.

If ,c jτ ατ> in any coordinate direction j , or m critr r> , cell c is flagged to be refined.

If ,c jτ βτ≤ in all coordinate directions and m critr r≤ for all the cells refined from the

same parent cell, the related child cells will be coarsened.

α and β are the control parameters and in this study were taken to be 1.0 and 0.2 by

preliminary tests, respectively, and 51.0 10critr −= × was the critical value, which was found to

be adequate. τ is the standard deviation calculated as

1/ 2

321,3

1 1

N

c jNj c

τ τ= =

⎛ ⎞= ⎜ ⎟⎝ ⎠∑∑ (4.4)

where N is the total number of the active cells. In addition, the difference in the refinement

levels at the cell interface is limited to one to ensure a smooth transition of AMR.

Engine simulations usually need to deal with moving boundaries such as moving pistons and

valves. In the KIVA code, inner grid points are rezoned to preserve grid quality after each

Lagrangian movement of the moving boundaries. The rezoning of general unstructured

meshes can be accomplished by solving the Laplace equation. To avoid the difficulty of

rezoning “hanging” nodes in the locally refined region, only the initial mesh (level 0) is

92

rezoned. The new locations of the refined cells can be determined by linear interpolation of

the locations of the updated initial mesh.

The local refinement introduces the coarse-fine cell interface that complicates the numerical

schemes for flux calculation. Fluxing schemes in the KIVA code must be modified to

preserve the robustness and accuracy of the solver. The governing equations of the gas phase

have been described in Chapter 2. For convenient reference, the same set of the governing

equations is given here in integral forms with k ε− turbulence model. The equations for

conservation of mass for species m are

( )m

c smc m m mlV s V V

D dV D d dV dVDt

ρρ ρ ρ ρ δρ

⎡ ⎤= ∇ + +⎢ ⎥

⎣ ⎦∫ ∫ ∫ ∫i A (4.5)

where ,m

ρ ρ are the density of species m and the total density, respectively, ,c sm mρ ρ are the

chemical and spray source terms, respectively, cD is the gas diffusion coefficient, and mlδ is

the Dirac delta function in which subscript l corresponds to a liquid fuel in a

multicomponent fuel injection. The equations for conservation of mass and momentum are

s

V V

D dV dVDt

ρ ρ=∫ ∫ (4.6)

and

02

1 23

s

V s s V V

D dV p A k d d dV dVDt a

ρ ρ ρ⎡ ⎤= − + + + +⎢ ⎥⎣ ⎦∫ ∫ ∫ ∫ ∫iu A σ A F g (4.7)

where u is the velocity vector, a is a pressure gradient scaling parameter for low Mach flow,

p is the pressure, k is the turbulent kinetic energy, 0A is a switch for turbulent or laminar

93

flow, σ is the viscous stress tensor, sF is the spray momentum contribution to the gas phase,

and g is the gravity vector. The energy equation is based on the specific internal energy as

0

0

(1 ) : ( )

mc mV V V s

m

c s

V V V

D IdV p dV A dV K T D h dDt

A dV Q dV Q dV

ρρ ρρ

ρε

⎡ ⎤= − ∇ + − ∇ + ∇ + ∇ +⎢ ⎥

⎣ ⎦

+ +

∑∫ ∫ ∫ ∫

∫ ∫ ∫

i iu σ u A(4.8)

where I is the specific internal energy, K is the thermal conductivity, T is the temperature,

mh is the specific enthalpy, ε is the turbulent dissipation rate, and ,c sQ Q are the energy

source terms from chemical reactions and spray, respectively. Turbulence is modeled by the

k ε− turbulence equations

2 ( )3 Pr

s

V V V s V Vk

D kdV k dV dV k d dV W dVDt

μρ ρ ρε⎡ ⎤

= − ∇ + ∇ + ∇ − +⎢ ⎥⎣ ⎦

∫ ∫ ∫ ∫ ∫ ∫i iu σ : u A , (4.9)

1 2 2( )Pr

st t sV V s V

D dV C dV d C c c W dVDt k ε

ε

μ ερε ρε ε ρε⎡ ⎤

⎡ ⎤= − ∇ + ∇ + ∇ − +⎢ ⎥ ⎣ ⎦⎣ ⎦

∫ ∫ ∫ ∫i iu A σ : u (4.10)

where μ is the viscosity coefficient, Prk and Prε are the Prandtl numbers, sW accounts for the

rate of work done by turbulent eddies to disperse spray droplets, 2,c cε ε are the model

constants, and 1tC and 2tC take on different forms for the standard k ε− or the RNG

(Renormalization Group) k ε− models (Amsden et al., 1989; Torres and Trujillo, 2006).

Due to the use of AMR, the calculation of diffusive fluxes needs to be modified from its

original algorithm to account for the coarse-fine mesh interface introduced by AMR. Terms

bearing a format of s

Q d∇∫ Ai in the gas phase equations are approximated by using the mid-

point quadrature rule,

94

( ) ffsf

Q d Q∇ ≈ ∇∑∫ A Ai i (4.11)

where subscript f represents a cell face. For a conventional face f in Figure 4.2 (a), the

term ( ) ffQ∇ iA is approximated by

( ) , 1,2 1 2 4,3 4 3( ) ( ) ( )f c cn c cnfQ a Q Q a Q Q a Q Q∇ = − + − + −iA (4.12)

Q3, X3

Q2, X2

Q4, X4Qcn, Xcn

Qc, Xc

Q1, X1

fA

Q3, X3

Q2, X2

Q4, X4Qcn, Xcn

Qc, Xc

Q1, X1

fA

C1 (l-1)

C2 (l)

C3 (l)

(a) Conventional interface (3-D) (b) Coarse-fine interface (2-D)

Figure 4.2. Geometric arrangement of the gradient calculation at an interface f.

where ,c cnQ Q are the cell-centered values of cell c and cell cn , 1 2 3 4, , ,Q Q Q Q are the edge-

centered values of edges 1, 2, 3 and 4. The edge-centered values are obtained by averaging

the cell-centered values of all cells that share the edge. ,c cna , 1,2a , and 4,3a are the geometric

coefficients and determined by solving a system of 3 3× equation

, 1,2 1 2 4,3 4 3( ) ( ) ( )c cn c cn fa a a− + − + − =x x x x x x A (4.13)

where ,c cnx x are the cell centers on either side of the face f , 1 2 3 4, , ,x x x x are the edge

centers of four edges bounding the face f , and fA is the face area vector equal to ˆf A=A n

where n̂ is the outward unit normal vector and A is the area of the face. At the coarse-fine

interface, Eq. (4.11) is applied to the l -level child cells (cells 2 3,c c in Figure 4.2 (b)). When

95

calculating the edge-centered values iQ (i.e., the dashed red edge in the cell 2c ), the

contribution from the ( )1l − -level cell (i.e., the cell 1c ) across the interface is achieved by

attaching the edge to the ( )1l − -level cell which is carried out in the refinement process. The

term Q∇ iA for the face of the ( )1l − -level cell becomes

( ) ( )1 ,l h lh

Q Q−

∇ ≈ − ∇∑A Ai i (4.14)

where ( ) ,h lQ∇ Ai is the value for one of the l -level cells.

The viscous stress tensor term s

dA∫ iσ in the momentum equations is approximated by

summing over all the cell faces

f fsf

dA ≈∑∫ i iσ σ A (4.15)

where fσ is the viscous stress tensor,

( ) 23

Tf t tμ μ⎡ ⎤= ∇ + ∇ − ∇⎣ ⎦σ u u uIi (4.16)

with tμ is the viscosity coefficient, superscript T indicates tensor transpose, and I is the

unit tensor. One must find the value of fσ and thus the velocity gradients ( ) fu∇ at each cell

face. For a velocity component u and a face f in Figure 4.2 (a), the gradients u∇ satisfies

the following condition

( )Tu∇ =X Δu (4.17)

where the matrix X is defined by

96

12 12 12

43 43 43

ccn ccn ccnx y zx y zx y z

⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠

X , (4.18)

and the right-hand side vector Δu has

1 2

4 3

c cnu uu uu u

−⎛ ⎞⎜ ⎟= −⎜ ⎟⎜ ⎟−⎝ ⎠

Δu . (4.19)

Then the components of the gradients u∇ in each coordinate direction can be determined as

follows,

( ) ( )( ) ( )( ) ( )

1

1

1

x

y

z

u u

u u

u u

−

−

−

∇ = ∇ =

∇ = ∇ =

∇ = ∇ =

i i X Δu

j j X Δu

k k X Δu

i i

i i

i i

(4.20)

where , ,i j k are the unit vectors in , , and x y z coordinate directions, respectively. The terms

1−ii X , 1−ij X and 1−ik X correspond to the 1st, 2nd and 3rd row of the matrix 1−X , respectively,

and can readily be obtained from Eq. (4.13). The gradients of other velocity components are

calculated in a similar way. Then, the viscous diffusion term f fiσ A for the face of a l -level

cell can be determined at the interface. For the face of the ( )1l − -level cell across the

interface, this term is obtained by summing the terms for all the l -level cells as

( ) ( )1 ,f f f fl h l

h−≈ −∑σ A σ Αi i (4.21)

where h represents a l -level cell at the interface.

97

One also needs to calculate the viscous dissipation of the mean flow kinetic energy

V

dV∇∫ σ : u since both tensors σ and∇u are evaluated at the cell center instead of the cell

faces. Let u be a component of velocityu . Then u∇ at the cell center is approximated as a

cellwise constant using the divergence theorem as

( )1, , 12

1 1 1 1f f c l c l fV S

f fu udV ud u u u

V V V V −∇ ≈ ∇ = ≈ ≈ +∑ ∑∫ ∫ A A A (4.22)

where fu is the face center velocity, and ,c lu , , 1c lu − are the cell-centered values at the levels l

and ( )1l − , respectively. At the coarse-fine interface, for an l -level cell, fu is simply

approximated by the average of the cell-center values ,c lu and , 1c lu − . For the ( )1l − -level

cell across the interface, ,c lu is the average of the cell-centered values of all the l -level cells

sharing the interface. After the gradient terms are determined at the cell center, σ can be

determined accordingly.

The collocation of pressure and velocity can cause parasitic pressure oscillations due to

velocity-pressure decoupling (Ferziger and Peric, 2002). To reduce the oscillations, the

Rhie-Chow pressure correction method (Tsui and Pan, 2006) is used when computing the

face volume change for the pressure solution in the Lagrangian phase. Using this technique,

the face volume change ( ) fu Ai is modified by pressure gradients according to the following

expression:

[ ]{ }121

2

( ) ( ) ( ) ( ) ( )( )

oLLf f f c cn f

c cn

t p p pρ ρΔ

= − ∇ − ∇ + ∇+

u A u A Ai i i (4.23)

98

where ( ) oLfu Ai is calculated from the original face volume change equation that is derived

from the momentum balance for control volume centered around the face f (Amsden et al.,

1989; Torres and Trujillo, 2006). ( ) ,( )c cnp p∇ ∇ are the cell pressure gradients and are

calculated using the divergence theorem

( )1, , 12

1 1 1 1f f c l cn l fV S

f fp pdV pd p p p

V V V V −∇ ≈ ∇ = ≈ ≈ +∑ ∑∫ ∫ A A A (4.24)

where ,c lp and , 1cn lp − are the cell-centered velocities. ( ) fp∇ is the pressure gradient at the

cell face and is determined by using the method in Eq. (4.12). ,c lp and , 1cn lp − are determined

in the same way as ,c lu , and , 1c lu − in Eq. (4.22). The face volume change for the ( )1l − -level

cell across the interface is then summed over all the l -level cells on the other side of the

interface as

( ) ( ),1f h fl l

L L

hu u

−

≈ −∑A Ai i . (4.25)

The term ( )Lfu Ai is used to calculate the Lagrangian cell volume LV that appears in the

pressure iteration as

( )L n Lf

fV V t= + Δ ∑ u Ai (4.26)

where nV is the cell volume at previous time step n and tΔ is the main computational time

step. The term ( )Lfu Ai will also be used to calculate convective face volume change when

the face moves from the Lagrangian position to the final position corresponding to the new

time step.

99

The calculation of convective fluxes also needs to consider the presence of the coarse-fine

interface. In the transient engine simulation, after the mesh moves with the fluid in the

Lagrangian phase, the mesh is rezoned to the new location which leads to the convective

transport of the flow fields due to the relative movement of the mesh to the frozen fluid. The

convective fluxes in the Eulerian phase are explicitly subcycled within the main

computational time step to save computer time. The number of subcycles is determined

by /sc cn t t= Δ Δ where tΔ is the main computational time step, ctΔ is the convective time step

which satisfies the Courant condition and will be determined later. At each subcycle v , the

face volume change fVδ associated with the cell face f is calculated by considering the total

face volume change from the Lagrangian position to the final position after rezoning as

( )1,l l l

Ln n cf f c f

tV V tt

δ δ + Δ= −Δ

Δu Ai (4.27)

where 1,l

n nfVδ + is the total volume change of the face f of the l -level cell from the location at

time n to the final location at time 1n + . The 1lf

Vδ−

for the ( )1l − -level cell at the interface

is obtained by taking the negative of the sum of lf

Vδ for the corresponding l -level cells. If

fVδ is positive, the face movement leads to the cell volume expansion. If it is negative, the

face movement leads to the cell volume compression.

The flux through a conventional face f into a cell c as shown in Figure 4.3 (a) is calculated

by

( ) ( ) ( )1 1

l

v v vfc c f

qV qV q Vρ ρ ρ δ− −= + (4.28)

100

face f

cncno

i i ic

face fo xLxf

C1 (l-1)C2 (l)

C3 (l)

(a) Conventional interface (3-D) (b) Coarse-fine interface (2-D)

δVf

Figure 4.3. Computational stencil for gradient calculation.

where ρ is the density, V is the cell volume, and q represents the cell-centered quantity

being evaluated such as velocity components, and specific internal energy. The quantity

( ) fqρ at each subcycle is determined by using a quasi-second-order upwind (QSOU)

scheme

( ) ( ) ( ) 1nn

n

ff cf c

c

Vd qq q

ds Vδρ

ρ ρ⎛ ⎞⎜ ⎟= + − −⎜ ⎟⎝ ⎠

x x (4.29)

where ( )d qdsρ

is the slopes in each cell computed by using a strong monotone upwind

scheme modified from Van Leer’s upwind scheme (Van Leer, 1979) as

( ) ( ) ( )( ) ( ) ( ) ( ) ( )sign min ,n n no

n

n n no

c c c c

c cc c c c

q q q qd qq q

ds

ρ ρ ρ ρρρ ρ

⎛ ⎞− −⎜ ⎟= − ⎜ ⎟− −⎜ ⎟⎝ ⎠

x x x x (4.30)

where , ,n noc c cx x x represent the cell-centered locations at the new time level while fx is the

face center. Lx is the Lagrangian position of the face. ( )d qdsρ

is zero if ( ) ( )nc c

q qρ ρ− and

( ) ( )n noc c

q qρ ρ− have different signs. For ( )d qdsρ

of the ( )1l − -level cell (i.e., 1c in Figure

101

4.3 (b)), ( )nc

qρ and ncx are the average values of all the l -level cells (i.e., 2c and 3c ) on the

other side. Only the faces from the l -level cells will be looped over to avoid a duplicate flux,

and ( )d qdsρ

will be determined for every face based on its associated volume change fVδ .

Eq. (4.28) is then applied to the l -level cells. The flux for the ( )1l − -level cell is given by

( ) ( ) ( ),1 1 ,

1 1

h ll l h l

v v vff f f

hqV qV q Vρ ρ ρ δ

− −

− −= −∑ (4.31)

where h∑ means summation over all the l -level cells at the interface.

The new time step 1nt +Δ is the minimum among an array of time step constraints (Amsden et

al., 1989). Two constraints need to be re-evaluated on the refined cells resulting from AMR

due to the coarse-fine interface and cell size change. First, the time step must be restricted to

limit the cell distortion due to mesh movement in the Lagrangian phase. This constraint is

enforced by a time step

( ) ( )21 1

1 2 3 2 1 3 3 1 23 9

min2 3

n rstrst c

ii ii ij i j i j i j ij i j

ftS S S S S S S Sε ε ε

Δ =⎡ ⎤+ − + +⎣ ⎦

. (4.32)

where rstf is 13 , ijS is the rate of strain tensor, lijε is the alternating tensor, and the repeated

indices imply Eisenstein summation. The gradient u∇ in the rate of strain tensor ijS is

calculated using Eq. (4.22). Second, the convection time step must reflect the cell size

change resulting from the refinement/coarsening. The convection time step is limited by

1 minn nc c c f

f

Vt C tVδ

−Δ ≤ Δ (4.33)

102

where cC is a constant 0.2, V is the cell volume, 1nct−Δ is the previous time step, and fVδ is

the face volume change. Unfortunately, fVδ is not available for the newly refined cells

using AMR since this term is calculated in the rezoning stage whereas Eq. (4.33) is

calculated at the beginning of each computational cycle. For newly refined cells, nctΔ can be

approximated by using geometrical relationships. For a child cell, its volume is

approximately one-eighth of its parent’s volume, and its face volume change is

approximately one-fourth of that of its parent cell. The convection timestep of each child cell

can be taken as half of the timestep for the parent cell. During the cell coarsening, the face

volume change fVδ for the re-activated parent cell can be approximated from its child cells.

Then Eq. (4.33) can be directly applied.

4.3 Parallelization

Parallelization of KIVA-4 was performed using the Message Passing Interface (MPI) library

for use in distributed memory machines. Before a parallel computation starts, a

computational mesh (domain) must be decomposed into all processors. Domain

decomposition influences work load balance, communication overhead, and thus overall

parallel performance. In engine simulation, domain decomposition is complicated by two

major factors. First, cells and nodes can become de-activated and/or activated during the

calculation. For instance, the entire port can be re-activated or de-activated after a valve is

opened or closed. Piston motion also causes layers of cells to be de-activated or re-activated.

These transient features can cause the initially balanced workload to become unbalanced.

103

Second, the modeling of spray particles can also cause the imbalance of workload and

communication overhead. If the spray particle is directly associated with the processor that

owns the cell where the spray particle is located, the parallel implementation can be

simplified but the workload may not be balanced. On the other hand, if the particles are

distributed equally among processors, the implementation will become difficult and the

communication overhead will be increased.

The work on parallelizing KIVA-4 was a collaborative effort with Los Alamos National

Laboratory (LANL) (Li et al., 2008; Torres et al., 2009). The current study particularly

contributed to the parallelization and performance for complex engine geometries. The focus

of the work presented in this report is on parallelizing AMR for transient engine spray

simulation (Li and Kong, 2009a).

To preserve the load balance, it is important to partition the domain such that each processor

has the same number of active cells. The current strategy is a static partitioning in which an

equal number of parent cells are assigned to each processor and used throughout the

calculation. For spray particles, to minimize communication overhead, the current

implementation is to associate each spray particle with the processor that owns the cell within

which the spray particle resides. Child cells are tied to the spray particles. Once the parent

cells are partitioned, the child cells are assigned to the same processor as their parent cell.

The partitioning of the parent cells is accomplished by using a constrained form of mesh

partitioning in METIS (Karypis and Kumar, 1998a). METIS is a software package for

104

partitioning unstructured graphs, partitioning meshes, and computing fill-reducing ordering

of sparse matrices.

In the current partitioning, an unstructured mesh is first converted into a graph in which all

cells become vertices and the faces between the cells become edges. The edges in the graph

can be assigned weights to form different partitions. In reciprocating engines, a

computational domain experiences periodic compression and expansion in a vertical

direction. In order to obtain a balanced computational load, a mesh is generally grouped

vertically into different processors. This can be achieved by assigning large weights to

vertically linked edges in the graph format. Figure 4.4 shows an example of the domain

divisions of three geometries for computations using four processors using the vertical

partition strategy only. These geometries are relatively simple, in particular, without port

deactivation and re-activation. For engine geometry with port de-activation/re-activation, a

single strategy of vertically partitioning the domain may assign processors a disproportional

number of cells from the port region which will cause load imbalances. This can partially be

addressed by partitioning different portions of the domain separately based on a hypergraph

partition in the hMETIS software package (Karypis and Kumar, 1998b). This second

strategy still needs to combine with a dynamic repartitioning strategy to obtain a balanced

load in the whole computational cycle (Torres et al., 2009).

A parallel computation consists of the following steps. First, domain partitioning assigns

each processor a subset of the total cells in the mesh. A processor owns all cells that are

assigned to it and all nodes that constitute the cells. Therefore, a node can be owned by

105

multiple processors. Then, each processor solves the hydrodynamic equations on its set of

cells. Cells and nodes around the processor boundary need the information of their neighbors

that are owned by other processors. However, different processors usually do not share the

memory in a distributed-memory computer cluster. Thus, information on cells and nodes

around the processor boundary needs to be built and exchanged during the calculation in

order to solve the equations correctly. Therefore, it is necessary to establish neighboring

cells and nodes of each processor so that the information can be exchanged between

processors, which is established in a so-called communication initialization.

Figure 4.4. Domain partition for four processors using METIS.

Communication initialization includes assigning cells and nodes and their associated

properties to each processor, establishing neighboring cells and nodes, and constructing the

communication arrays. This initialization is completed before the hydrodynamic calculation

begins. Data exchange is performed during the hydrodynamic calculations by calling

different communication subroutines which will be described shortly. The assignment of

106

cells and nodes to each processor can be readily accomplished based on the partition file

from the domain decomposition.

The neighboring cells of a processor are defined as the cells that do not belong to the

processor but touch any of the nodes that belong to it. Similarly, the neighboring nodes of a

processor are defined as the nodes of the neighboring cells of the processor but exclude those

nodes that are owned by the processor itself. In the code, the neighboring cells are used to

exchange cell-based properties and face properties during the calculation. The neighboring

nodes are used to exchange coordinates and velocities of the nodes. The information on the

neighboring cells and nodes, which usually includes lists of cells and nodes that need to be

sent to and received from neighboring processors, are saved in the communication arrays.

Different communication subroutines are developed to exchange different types of data in the

neighboring cells and nodes for reduced communication cost. For instance, in the present

code, subroutine communicate is designed to communicate floating point cell-based

properties. Subroutine communicatei is designed to communicate integer cell-based

properties. Subroutine communicatev is designed to communicate floating point node-based

properties. Subroutine communicatederivativese is designed to communicate derivates of

cell-based properties at a face interface that lies on the processor boundary. This subroutine

is called to exchange the derivatives during the convection calculations. Subroutine

communicatespd is designed to communicate cell-based species densities. The construction

of these subroutines will enhance the efficiency of code implementation as well as parallel

performance by reducing the searching cost.

107

AMR is performed after the initial communication setup. The communication re-

initialization is required due to the changes of cells, nodes, connectivity, and properties.

Newly created child cells and nodes are simply assigned to the same processor as their parent

cells as mentioned above. To reduce the re-initialization cost, however, the process of

finding neighboring cells and nodes and constructing the communication arrays requires

special attention, as will be described in the following.

Due to the hanging node structure on the local mesh interface (Figure 4.1 (b)), it is necessary

to loop through both parent cells and child cells to find the neighboring cells and nodes at

each re-initialization. The search cost can be reduced by considering the cells and nodes on

the processor boundaries only. The efficiency can be further improved if the neighbors do

not need to be re-searched when a new refinement does not occur on any of the processor

boundaries. One can also refine all the cells on the processor boundaries at first AMR so that

the neighbors can be set up one time only. However, this strategy may increase unnecessary

communication and can cause more communication overhead.

The communication arrays depend upon both the total numbers of cells and nodes owned by

each processor and the numbers of its neighboring cells and nodes. The arrays can stay

unchanged if AMR is merely an activity of cell coarsening and/or re-activation of the de-

activated child cells in which both the cells and nodes owned by each processor and its

neighboring cells and nodes do not change. However, only the local addresses of the

108

neighbors in the arrays need to be updated if a new refinement does not occur on any of the

processor boundaries in which the numbers of the neighboring cells and nodes do not change.

In fact, the communication re-initialization is generally costly in computer time if it is

performed frequently. Therefore, the strategy is to restrict the frequency of performing AMR,

e.g., every 10 computational timesteps. The solution accuracy was sacrificed slightly to

obtain better parallel performance. This strategy may be carefully tuned to obtain a good

trade-off between accuracy and speed-up. Other potential strategies have been considered

for future implementations such as repartitioning the cells to obtain a better load balancing

among processors at each adaptation timestep. Furthermore, parallel AMR may be improved

by executing the mesh refinement/coarsening on each involved processor instead of

executing it on root process only, as is the case in the current implementation.

4.4 Gasoline Spray Simulation Using AMR

The baseline code used in this part of work is parallel KIVA-4. Fuel spray atomization was

simulated using the Taylor Analogy Breakup (TAB) model (O’Rourke and Amsden, 1987)

that was available in KIVA-4. The TAB model is known to predict gasoline spray breakup

reasonably well. KH-RT models along with a nozzle flow model were also implemented into

the parallel KIVA4 for modeling diesel spray atomization. Simulations using the KH-RT

models and AMR were performed and validated in later work (Kolakaluri et al., 2009).

Other spray sub-models included the models for collision, turbulence dispersion,

vaporization, and wall film to account for related phenomena. Combustion was not

109

considered here. The standard k ε− model was used to model the effects of turbulence on

mean flow and spray. The law-of-the-wall boundary conditions were used to treat turbulence

and heat transfer near walls. The fixed temperature condition was used for wall temperature.

The initial spray conditions were determined from the injector geometry and injection

conditions. The serial AMR implementation was first validated followed by the validation of

parallel AMR. Then the parallel performance was tested on a number of engine geometries.

Verification of AMR implementation

The present AMR method was applied to simulate gasoline sprays (Li and Kong, 2009b).

The engine had a cylindrical combustion chamber with the bore and stroke equal to 10 cm

and 10 cm, respectively. The engine speed was 4,000 rev/min. The spray was injected at a

constant rate of 135 m/s into the chamber from the center of the cylinder head. The initial

drop diameter was 200 microns and the initial drop temperature was 301 K. The fuel was

iso-octane and the total injected fuel was 3.6 mg. The ambient was air with an initial

temperature of 301 K and pressure of 1 bar. The initial meshes were based on the Cartesian

O-grid type mesh. The simulation started from 180 ATDC when the piston started to move

upward, and the fuel was injected also at 180 ATDC for a duration of 30 crank angle degrees.

It is expected that a coarse mesh using AMR should give the same results as the uniformly

fine mesh but with reduced computer time. Therefore, results using the uniformly fine mesh

will serve as the baseline for AMR verification. Notice that “coarse” and “fine” are only

relative terms for describing the mesh density. The coarsest mesh size was 5 mm. To meet

the large void fraction condition of the Lagrangian method, AMR was performed up to 2

110

levels in this study. Thus, the comparable finest mesh size was 1.25 mm on average.

Simulations were performed for the following mesh conditions: 5 mm, 2.5 mm, 1.25 mm, 5

mm with one-level refinement (5mm+L1), and 5 mm with two-level refinement (5mm+L2).

Due to the stochastic nature of spray modeling, comparisons using representative statistical

data of spray would not be feasible. Therefore, the liquid and vapor penetrations, vapor mass

fraction, and spray structure were chosen as parameters for comparison. The liquid

penetration was defined as the distance from the nozzle orifice to a position which

corresponds to 95% of the integrated amount of liquid drops from the nozzle orifice. The

vapor penetration was the axial distance from the nozzle orifice to a position with the vapor

mass fraction greater than 0.01.

0.01.02.03.04.05.06.07.08.09.0

10.0

180 190 200 210 220 230

Crank angle (ATDC)

Liqu

id P

enet

ratio

n (c

m)

1.25mm2.5mm5.0mm5.0mm + L15.0mm + L2

(a) Liquid penetration (cm)

111

0.01.02.03.04.05.06.07.08.09.0

10.0

180 190 200 210 220 230

Crank angle (ATDC)

Vap

or P

enet

ratio

n (c

m)

1.25mm2.5mm5.0mm5.0mm + L15.0mm + L2

(b) Vapor penetration (cm)

Figure 4.5. Comparison of liquid and vapor penetrations using different meshes with respective refinement levels.

Table 4.1. Comparison of the computer time for different mesh sizes.

mesh size Δx 1.25 mm 2.5mm 5.0mm 5.0mm+L1 5.0mm+L2 Time (min) 4770.9 311.3 15.4 88.5 467.2 % (based on the 1.25 mm mesh case)

100.0 6.52 0.32 1.85 9.79

Figure 4.5 shows the comparisons of liquid and vapor penetrations for the five cases studied.

The 5mm+L1 case produced similar penetrations to those using 2.5 mm mesh size. The

5mm+L2 case produced comparable penetrations to those using 1.25 mm mesh size. It can

be seen that the case using 5 mm mesh size without AMR produced the shortest penetrations.

As the mesh density increased, so were the penetrations because the increased grid resolution

112

improved the prediction in momentum exchange from the spray to the gas phase resulting in

smaller relative velocities and thus longer penetrations, as can be seen from the maximum

gas velocity prediction in Figure 4.6. Table 4.1 shows the comparisons of computer times

for the cases studied for computations from 180 to 360 ATDC (i.e., piston at top-dead-center).

It is seen that the time saving for both level 1 and level 2 refinements are significant

compared to the equivalent fine mesh cases for the same level of solution accuracy. This time

saving can be justified by total active cell numbers that participated in computations in each

mesh case as shown in Figure 4.7. The total active cell numbers in the 5mm+L1 case

amounted to only 28% of the active cells using 2.5 mm mesh size at the early stage of the

calculations and only to 50% of those by the end of the calculations. There were much fewer

active cells participating in computation in the 5mm+L2 case compared to the 1.25 mm case.

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

180 190 200 210 220 230

Crank angle (ATDC)

Max

. Gas

Vel

ocity

(m/s)

1.25mm2.5mm5.0mm5.0mm + L15.0mm + L2

Figure 4.6. Comparison of the maximum gas velocity for different mesh refinement

conditions.

113

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E+06

180 200 220 240 260 280 300 320 340 360

Crank angle (ATDC)

Tot

al A

ctiv

e C

ell N

umbe

r

1.25mm2.5mm5.0mm5.0mm + L15.0mm + L2

1.25mm

2.5mm

5.0mm

5.0mm + L2

5.0mm + L1

Figure 4.7. Total active cell numbers as a function of crank angle for different mesh

refinement conditions.

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E+06

180 200 220 240 260 280 300 320 340 360

Crank angle (ATDC)

Tot

al C

ell N

umbe

r

1.25mm2.5mm5.0mm5.0mm + L15.0mm + L2

1.25mm

2.5mm

5.0mm

5.0mm + L2

5.0mm + L1

Figure 4.8. Total cell numbers as a function of crank angle for different mesh refinement

conditions.

114

The fluctuations seen on the curves were results of the cell de-activation due to the snapping

procedure that was used to de-active or re-active a layer of cells to preserve grid quality when

a computational domain was compressed or expanded. Figure 4.8 shows the total cells

(active plus inactive) for the different mesh sizes. It can still be seen that the AMR cases

entailed significantly fewer cells compared to the equivalent fine mesh cases.

Figure 4.9. Comparison of the spray pattern and fuel vapor on a cutplane along the spray axis

at 205 ATDC.

Figure 4.9 shows the fuel vapor distribution and spray pattern for the five cases on a

cutplane along the spray axis at 205 ATDC with all the drops projected onto the cutplane.

Similar fuel vapor distributions can be observed between the “5mm+L1” case and the “2.5

mm” case. The same is true for the results between the “5mm+L2” case and the “1.25 mm”

case. In the “5.0 mm” case, fuel vapor was distributed in a wider region than in other cases

due to over-estimated diffusion resulting from the coarse grid size. Due to the insufficient

spatial resolution, the “5.0 mm” case produced the so-called “clover leaf” artifact (Figure

115

4.10 (c) bottom) (Schmidt and Rutland, 2000; Hieber, 2001). The artifact is a result of drop

collisions that move the new drops towards the cell centers around the injector since the

drops moving 90° apart in the same cell tended to move towards the cell center due to the

large relative velocities between the drops. AMR refined all the cells around the injector and

thus distributed drops into more cells, alleviated this artifact, and produced the pattern similar

to that using the fine mesh as seen in Figure 4.10 (a,b,d,e).

Figure 4.10. Comparison of the spray pattern on a cutplane along the spray axis (top) and at

3.0 cm above the piston surface (bottom) at 205 ATDC.

It can be seen that the spray simulation is dependent on the base mesh if AMR is not used.

To further demonstrate the benefits of using AMR, simulations were performed in the same

domain by injecting spray at a 40° angle from the vertical axis. The ambient pressure was 3

bars. Simulations were performed on the 1.25 mm, 5.0 mm, and 5.0mm+L2 refinement.

116

Figure 4.11 shows the comparisons of spray structure and vapor contour at 1 ms after

injection. The coarse mesh without AMR (5.0 mm) predicted a shorter penetration and

higher vapor diffusion than those using the fine mesh or the coarse mesh with AMR. It is

also seen that the coarse mesh without AMR predicted asymmetric shear stresses on the top

and bottom of the spray tip causing an asymmetric spray structure. The results using the L2

refinement are comparable to those using the 1.25 mm mesh.

Figure 4.11. Comparison of the spray pattern and fuel vapor for injection with a slanted

angle.

The above comparisons indicate that the AMR method can produce comparable results to

those using the fine mesh but with much less computer time (less than 10% of the time for

the fine mesh). In particular, AMR refined all cells that contained large drops or have high

vapor gradients. AMR can also effectively alleviate the “clover leaf” artifact and other

artifacts in the Cartesian mesh.

Verification of parallel AMR implementation

Three engine geometries were used to verify and evaluate the parallel AMR implementation

(Li and Kong, 2009b). The simulation conditions are listed in Table 4.2 for these three

117

geometries including a cylindrical chamber, a 2-valve model, and a DI gasoline engine. To

test the effects of the spray on the load balance and communication overhead, both one jet

and six jets were simulated. Two types of meshes (coarse and fine) were used to evaluate the

effect of mesh size on the parallel performance. Cell numbers listed in Table 4.2 were the

Table 4.2. Simulation Conditions.

Engine Model

Cylindrical chamber

2-valve engine DI Gasoline engine

Bore (cm) 10 14 5 Stroke (cm) 10 10 10 Piston and dome Flat Flat Complex shapes Cell numbers (coarse) 15,225 16,000 55,950 Cell numbers (fine) 124,200 128,000 193,500 Fuel Gasoline Gasoline Gasoline Number of nozzle holes 1 or 6 1 or 6 1 or 6 Start of injection (ATDC) 180 360 300 Injection duration (CAD) 30 10 40 Injection velocity (m/s) 135 # of comp. parcels 2000 Injection rate constant Initial droplet size 100 micron 430 micron 236 micron Initial droplet temp. (K) 301 331 301 Initial gas temp. (K) 300 800 300 Initial gas pressure (bar) 1 6 1 Engine speed (rpm) 4,000 1,000 1,500 Simulation duration (ATDC) 180 – 360 0 – 450 270 – 360

numbers of the total cells when the piston was at its bottom-dead-center position. The

geometries of the three different engines are shown in Figure 4.4. The parallel

implementation was verified by comparing the predicted results with and without parallel

computing. Since the stochastic particle technique was used for modeling the spray dynamics

118

and the numbers of parcels from processor to processor were not generally the same, the

resulting sets of random numbers associated with the parcels were not the same among

processors. As a result, the spray distribution may not be identical in a parallel run to that in

a serial run. Nonetheless, the comparisons of spray patterns are informative in assessing the

validity of the present implementation.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

180 190 200 210 220 230

Crank angle (ATDC)

Pene

trat

ion

(cm

)

no AMR (np=1)AMR (np=1)AMR (np=2)AMR (np=4)AMR (np=8)AMR (np=16)

Figure 4.12. Comparison of liquid penetrations for different numbers of processors.

Figure 4.12 shows the comparison of spray penetrations predicted by using different

numbers of processors for the cylindrical chamber with the grid size of 4 mm for AMR cases.

Results using 2 mm grid size without AMR are also shown for comparison. Only slight

differences in the penetration are seen among the different cases. Hence, it can be seen that

the parallel AMR can achieve the same level of accuracy as in the serial run. The spray

patterns at crank angle 205 ATDC are shown in Figure 4.13 that includes the mesh on the

central cutplane with all the drops in the plot. The slight differences in the spray patterns

119

among processors are thought to be due to the different sets of random numbers generated

from each processor as mentioned earlier. In fact, similar differences in spray patterns were

observed when simulations were performed using different numbers of processors without

using AMR (not shown here). Therefore, it can be seen that the parallel AMR was

implemented correctly.

Figure 4.13. Comparison of spray patterns at crank angle 205 ATDC using different numbers

of processors together with DAMR.

Figure 4.14 and Figure 4.15 show the comparisons in penetrations and spray patterns when

the 10 time-step control strategy was applied. It is seen that the penetrations are reasonably

close to the results that used the fine mesh without AMR. The same tests were also

performed on the other two geometries (i.e., 2-valve engine and DI gasoline engine) and

similar results were found. For brevity those results are not presented here.

120

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

180 190 200 210 220 230

Crank angle (ATDC)

Pene

trat

ion

(cm

)

no AMR (np=1)AMR (10-step, np=1)AMR (10-step, np=2)AMR (10-step, np=4)AMR (10-step, np=8)AMR (10-step, np=16)

Figure 4.14. Liquid penetration comparison with the 10 timestep control strategy.

Figure 4.15. Spray pattern comparison between the baseline serial run and parallel runs with

the 10-timestep control strategy.

Figure 4.16 shows the comparisons of the predicted spray structure with six jets using one

processor and four processors. The start of injection was 180 ATDC and the results shown

were at 310 ATDC. The top two pictures show the grids and fuel vapor contours on a cut-

121

plane together with all the fuel drops in the chamber. It is an artifact due to post-processing

that some drops appear to be out of bounds. The bottom two pictures show the spray

structures colored by the drop sizes. This geometry has irregular cells in the dome region

Figure 4.16. Spray pattern comparisons in the DI gasoline engine combustion chamber at

crank angle 310 ATDC. Note that the start of injection was 180 ATDC.

which leads to distorted cells in the squish region on the perimeter of the cylinder during

rezoning. Thus, a rezoning algorithm was implemented to alleviate the distortion of cells

near the squish perimeter. This algorithm requires 10 times more iterations and needs to

communicate neighboring cells during each iteration in a parallel run. Therefore, a parallel

run needs additional communication cost compared to a serial run when rezoning takes place.

122

Nevertheless, the results demonstrated that the present parallel implementation performed

properly for this geometry.

Since the goal of parallel computing was to reduce the elapsed time, the performance of the

parallel AMR was assessed by the speed-up and computational efficiency. The effects of

load imbalance and communication overhead can be assessed by comparing the speed-up and

efficiency. Furthermore, the cases with single-hole nozzle and 6-hole nozzle were designed

to test the effects of child cells on the load imbalance and communication overhead.

The speed-up S is defined as

1

N

TS T= (4.34)

where 1T , NT are the elapsed times of simulations by using 1 processor and N processors,

respectively. Ideally, the speed-up should scale linearly with the number of processors. In

reality, due to communication overhead and other factors, S is always lower than N and is

mostly dictated by the slowest processor. The computational efficiency E is defined as the

speed-up relative to the number of processors,

1

N

TSE N NT= = (4.35)

and indicates how effectively on average a processor is used for computing.

For the single-hole injection cases using AMR, Figure 4.17 shows the speed-up and

computational efficiency for the three geometries. The coarse meshes (see Table 4.2, and the

123

grid size is 4 mm for the cylindrical domain) were used. The speed-up was reasonable given

that the mesh sizes were relatively coarse and no combustion was involved. The DI gasoline

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18

number of processor

Spee

d-up

cylin. domaingasoline engine2-valve

0102030405060708090

100

0 2 4 6 8 10 12 14 16 18

number of processor

Com

puta

tiona

l Effi

cien

cy (%

)

cylin. domaingasoline engine2-valve

Figure 4.17. Speed-up and computational efficiency of the three geometries for single-hole

injection using the coarse mesh.

engine geometry case had the best speed-up partially because this geometry had the largest

number of grid points compared to the other two geometries. The computational efficiency

124

decreases as the number of processors increased due in part to the quality of partitioning with

regard to AMR and snapping. The decrease in efficiency beyond four processors was also

seen in the implementation by Bella et al. (2006) for computations without AMR.

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18number of processor

Spee

d-up

cylin. (6-jet)gasoline (6-jet)2-valve (6-jet)cylin. (1-jet)gasoline (1-jet)2-valve (1-jet)

0102030405060708090

100

0 2 4 6 8 10 12 14 16 18

number of processor

Com

puta

tiona

l Effi

cien

cy (%

) cylin. (6-jet)gasoline (6-jet)2-valve (6-jet)cylin. (1-jet)gasoline (1-jet)2-valve (1-jet)

Figure 4.18. Comparisons of speed-up and computational efficiency for single-hole and six-

hole injections using the coarse mesh.

125

Figure 4.18 shows the speed-up and computational efficiency for the 6-hole injections using

the coarse meshes. Compared to the single-jet cases, the six-jet cases had mixed influences

on the parallel performance for different geometries. For the cylindrical domain, a better

speed-up was obtained for the six-jet injection, indicating that the six jets may promote a

better partitioning of spray particles and the corresponding child cells among processors

during the AMR process. By contrast, the six-jet injection in the 2-valve case resulted in a

decrease in parallel performance. A possible reason for the performance decline may be

because the child cells were not well distributed due to strong in-cylinder flow motion

resulting from the opening and closing of the valves. Another possibility may be that the

root processor was over-loaded and thus slowed down the overall performance. In addition,

the 2-valve geometry suffers from extensive snapping due to the valve motion compared to

the other two geometries without valve motion. On the other hand, the results of the DI

gasoline engine geometry showed a similar performance for both the single-jet and the six-jet

cases.

The above tests show the scalability of parallel AMR with the number of processors for

different geometries. The parallel performance can also be influenced by the number of grid

points of a mesh through the ratio of computation to communication. Figure 4.19 shows the

variations of the speed-up and computational efficiency, respectively, for the coarse and fine

meshes (Table 4.2). The speed-up increased with the number of grid points in a domain.

This was due in part to the fact that sequential overheads remain relatively constant with the

126

increase in the number of grid points. Consequently, the fraction of computer time designated

for flow solution increased, resulting in a higher speed-up for the increased grid resolution.

0

2

4

6

8

10

12

0 2 4 6 8 10 12 14 16 18

number of processor

Spee

d-up

cylin. - coarse2-valve - coarsecylin. - fine2-valve - fine

0102030405060708090

100

0 2 4 6 8 10 12 14 16 18number of processor

Com

puta

tiona

l Effi

cien

cy (%

)

cylin. - coarse2-valve - coarsecylin. - fine2-valve - fine

Figure 4.19. Comparisons of speed-up and computational efficiency between coarse mesh

and fine mesh for using single-hole nozzle.

127

4.5 Gasoline Engine Simulation Using MPI-AMR

AMR was carried out on two realistic engine geometries: a 4-valve pent-roof engine and a 4-

vertical-valve (4VV) engine, as shown in Figure 4.20. The challenges of performing AMR in

these realistic geometries are twofold. First, irregular cells in the complex regions such as

valves and piston in unstructured meshes pose a difficulty in defining correct connectivity

data that are important to hydrodynamic calculations. Second, high aspect ratios in the

irregular cells can cause the solver to be unstable.

Figure 4.20. 3-D meshes of a 4-valve pent-roof engine and a 4-vertical-valve engine.

Compared to the 2-vertical-valve engine mesh, the 4-valve pent-roof mesh consists of both

irregular cells around the valves and cells with higher aspect ratios. The irregular cells

around the valves involve the definitions of boundary faces, edges and nodes that are critical

to the application of boundary conditions. Splitting of these cells makes it more difficult to

correctly define connectivity data due to possibly different scenarios of the cell splitting. The

(a) A 4-valve pent-roof engine (b) A 4-vertical-valve engine

128

high-aspect-ratio cells cause the solver to be unstable due to large convective fluxes in the

child cells during the rezoning phase. Different scenarios of cell splitting at the valve edges

were considered to fix the errors found related to the connectivity data. An alternative to

address the issue related to the connectivity would be to avoid cell splitting in one layer of

cells that touch the valve surfaces. The first approach was used in this study. The solver was

more stable by improving the calculations of the geometric coefficients used in the diffusion

calculations. This improvement results in more accurate calculations of the face volume

change ( ) fu Ai in Eq. (4.23) which is also used in the convective fluxing schemes.

Figure 4.21. Comparisons of fuel vapor mass fraction and temperature for 4-valve pent-roof

engine with and without AMR at 35 ATDC (SOI=10 ATDC).

The fuel in the 4-valve pent-roof engine was directly injected into the cylinder for testing the

code robustness. The fuel was injected at 135 m/s at 10 ATDC using a 6-hole nozzle with a

diameter of 200 micron. The injection duration was 48 CAD. The fuel mass was 0.07 g. The

engine speed was 1500 rpm. The bore and stroke were 9.2 cm and 8.5 cm, respectively. The

129

standard k ε− turbulence model was used along with standard spray models in KIVA-4 (i.e.,

TAB breakup model). The boundary conditions used the law-of-the-wall conditions. The

simulation started at -15 ATDC and ended at 400 ATDC. The total cell number was 38,392

when the piston was at bottom-dead-center. Figure 4.21 shows the comparisons of fuel

vapor mass fraction and temperature at 35 ATDC for the cases with and without AMR. It is

observed that AMR predicted better drop breakup due to the improved coupling from the

increased spatial resolution. Fuel vapor mass fractions in the AMR case were higher in a

wider region than those without AMR since the increased grid resolution in the AMR case

reduced over-estimated vapor diffusion which appeared in the case without AMR, in

particular, in the region right above the piston surface where the mesh was relatively coarse.

The higher vapor mass fraction was also a result of more vaporization which resulted in the

lower temperature in the AMR case as shown in the temperature plots. Figure 4.22 shows

the speed-up and computational efficiency, respectively. Under the current conditions, the

simulations were not as efficient as those in Table 4.2 with simple geometries. Uneven

particle distributions resulted from the valve motion, coarse mesh, and extensive valve

snapping could be part of the reasons that attributed to the deteriorated performance. Further

improvement is needed to address these issues.

The mesh for the 4-vertical-valve engine was an unstructured mesh generated by Ford Motor

Company using a grid generation package ICEM-CFD. As with the 4-valve pent-roof mesh,

many irregular cells had to be created around the valves due to the need of mesh generation

topology to accommodate the 4-vertical valves. The regions with irregular cells extended

from near the valves down to the piston surface. As with the 4-valve pent-roof mesh, the

130

splitting of cells around the valves and the piston bowl surface caused difficulties in correctly

defining connectivity data; and the high-aspect-ratio cells caused the hydrodynamic

calculations unstable. The improvements discussed in the case of 4-valve pent-roof mesh

were also used to fix these issues.

0

1

2

3

4

5

6

0 2 4 6 8 10

number of processor

Spee

d-up

No-AMRAMR

0

1020

30

40

50

6070

8090

100

0 2 4 6 8 10

number of processor

Com

puta

tion

al E

ffic

ienc

y (%

)

No-AMRAMR

Figure 4.22. Parallel performance for 4-valve pent-roof engine with and without AMR

(SOI=10 ATDC).

Figure 4.23. Comparisons of fuel vapor mass fraction and spray for 4-vertical-valve engine at

485 ATDC (SOI=460ATDC).

131

The engine bore was 103.75 mm and its stroke was 107.55 mm. The engine speed was 1,000

rpm. The fuel was injected at 135 m/s at 460 ATDC using a 6-hole nozzle with a hole

diameter of 170 micron. The injection duration was 50 CAD. The fuel mass was 0.05 g.

The total cell number was 81,176 when the piston was at its bottom-dead-center. The

calculations started at 100 ATDC and ended at 720 ATDC. The standard k ε− turbulent

model and spray models were used. The boundary conditions included the law-of-the-wall

conditions. Figure 4.23 presents the comparisons of fuel vapor mass fraction at 485 ATDC

for the cases with and without AMR. Due to the improved inter-phase momentum coupling

resulting from the increased spatial resolution, the penetrations using AMR were predicted

longer than those without AMR. As a result, more fuel vapor can be seen in the region with

spray. The speed-up in parallel tests was good under the current test conditions, as can be

seen in Figure 4.24.

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16 18number of processor

Spee

d-up

No AMRAMR

Figure 4.24. Speed-up for 4-vertical-valve engine with and without AMR (SOI=460ATDC).

132

In summary, the present parallel AMR algorithm was applied to simulate gasoline spray

dynamics in realistic engine geometries. Results using AMR are consistent with those in the

constant-volume chamber cases, i.e., AMR predicted long spray penetrations and could

resolve more detailed fuel vapor structures. Although detailed in-cylinder experimental

spray data are not available for model validation, it is anticipated that simulations using

parallel AMR can effectively provide more accurate and detailed spray and fuel-air mixture

distributions. Nonetheless, future simulations activating combustion chemistry can be

performed to further compare numerical results with engine combustion data.

133

5 CONCLUSIONS AND RECOMMENDATIONS

5.1 Conclusions

Two major tasks have been accomplished in order to develop predictive engine simulation

models based on the KIVA code. The first task was to implement the LES turbulence model

coupled with detailed chemistry to simulate diesel spray combustion. The present LES

adopted a one-equation dynamic structure model for the sub-grid scale stress tensor and the

gradient method for the sub-grid scalar fluxes. It was found that the present models

performed well in predicting the overall performance of engine combustion including the

cylinder pressure history, heat release rate data, and soot and NOx emissions trends with

respect to injection timing and EGR levels. The present LES approach could also predict the

unsteadiness and more detailed flow structures as compared to the RANS models. Therefore,

the current LES model can be further developed into an advanced engine simulation tool to

address issues such as cycle-to-cycle variations and to capture performance variations due to

the subtle change in engine operating conditions or geometrical designs.

The second task was to improve spray simulation by increasing the spatial resolution of the

spray region by using an adaptive mesh refinement (AMR) approach. AMR can be applied

to improve the phase coupling between the gas and liquid and thus improve overall spray

simulation. A grid embedding scheme was adopted in AMR to increase the spatial resolution

of a hexahedral mesh. The fine grid density was coarsened if this was not needed in the

spray region. These dynamic processes were controlled by using a criterion that incorporated

134

the normalized fuel mass and fuel vapor gradients. The grid refinement/coarsening were also

parallelized based on the MPI library in memory-distributed machines.

The parallel AMR algorithm was applied to simulate transient sprays in gasoline engines.

The present adaptive mesh refinement scheme was first shown to be able to produce results

with the same levels of accuracy as those using the uniformly fine mesh but with much less

computer time. Various spray injection conditions were tested in different geometries. In

general, the computations without valve motion or using a fine mesh could give better

parallel performance than those with valve motion or using a coarse mesh. Compared to the

single-jet injection, the six-jet injection had mixed influences on the parallel performance for

different geometries, which was considered to be related to the details of domain partitioning

and local mesh refinement in different geometries. The parallelization strategy, domain

partition, sprays, valve motion, and mesh density can all influence the final parallel

performance. Additionally, low quality cells such as irregular cells with the high aspect-

ratios in realistic engine geometries could seriously affect the robustness of the solver.

5.2 Recommendations

For the LES modeling, despite that the current sub-grid scale stress model seems to work

well in a wide range of engine applications, the sub-grid scale models for scalar fluxes,

turbulence-droplet interactions, and turbulence-chemistry interactions require further

investigation. An improved LES wall model may also be needed to improve momentum and

heat transfer modeling near walls using a RANS-type mesh. In addition, the RANS-based

135

spray sub-models may also need improvements in the context of LES flow field (Hu, 2008).

Further improvements of the sub-models and the inter-phase couplings will be needed to

reduce grid-dependent spray simulation.

The present AMR algorithm can be improved in parallel computation of complex engine

geometries. Specifically, local execution of AMR on each involved process and domain re-

partitioning following each adaptation may be needed to reduce communication overhead

and obtain more balanced computation. It is also important to enhance the robustness of the

solver in handling low quality cells that often come with meshes of realistic engine

geometries.

136

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